How did launched vehicles match speed of slingshot vehicle?

The original plan was to send a ship to mars with supplies. That ship explodes.

Then the Chinese offer to help. And the crew offers to sling shot back to mars.

The Chinese technology is good to get the pod up out of the earth's atmosphere. it's not good enough to go all the way to mars. It's a tiny ship. At best a one way trip to mars.

The crews ship doesn't have enough juice to land on mars. If they landed on earth then it would take too long for a relaunch. By using the sling shot they save time. But the trip to mars will end up starving the crew. Hence the food is sent to them instead.

Think of a relay race. The runner approaches with the baton. His successor runs alongside him for a while, matching the runner's speed, so that they can easily pass the baton.

That's basically it.

However, you have correctly added that this means that the supply ship would need to match the velocity of Hermes, which, as was established in the movie, needed a slingshot around Earth to achieve such a velocity.

However, do not ignore mass differences:
Acceleration is directly influenced by the mass of the accelerating object.

If the Hermes were to race against a ship that has half the mass; and they use the exact same thrust, the smaller ship will accelerate twice as fast. Or, in other words, it would take the smaller ship half the amount of time to accelerate to a given velocity, compared to how long it would take Hermes to reach that same velocity.

The supply ship was very small (relative to Hermes), and therefore could very quickly accelerate and decelerate in order to sync up with Hermes.

I can't remember if the supply ship was manned or not. Assuming it was not manned, this means that the supply ship does not need to limit its acceleration to the G force that a human body can withstand. An unmanned ship is therefore less constrained, and can accelerate at full blast (depending on the structural integrity of the ship)

Imagine the same relay race. The current runner, holding the baton, is an elephant. The successor, who needs to match the elephant's speed, is a cheetah. Obviously, the cheetah needs considerably less warm up time to accelerate and match the elephant's speed.

Compare this to the second relay team, which only consists of elephants. The successor, also an elephant, needs to start running much earlier than the cheetah, because he can only accelerate at a lower rate. He takes more time* so that he can accelerate up to the same velocity (= the runner's velocity), and only then is able to make the hand off.

*And the elephant also runs a longer distance than the cheetah while getting up to speed; which is a slightly more appropriate analogy for the Hermes/supply ship scenario, which was limited by the amount of Hermes' orbit being in lower Earth orbit.

That is a more accurate comparison of the Hermes and the supply ship; since their mass (and therefore means of acceleration) is very different.

Why not just send the supply ship then, since it is faster?

First of all, the supply ship is not faster (= higher max speed). It accelerates faster (= reaches max speed faster).
Think of it like a van and a Lamborghini driving on a highway at the maximum legal speed. The Lambo will reach the maximum speed much faster than the van; but once the van has reached maximum speed too; both cars will travel at the same speed. If you are making an incredibly long journey, the difference between travelling in the Lambo or in the van are negligible.

For the pedants: there is no max speed for any spaceship (other than the physical limitation of lightspeed) from a physics perspective. However, there are limitations as to travel speed from an engineering perspective. Most notably fuel costs, the risk of overshooting, and the ship hull's structural stability.

Secondly, while the supply ship is faster, it is not capable of sustaining human life for long enough, since the ship itself if too small. Yes, it carries the food supplies (and presumably oxygen and CO2 filters and the likes), but it is not a fully functional ship like Hermes is.

For the same reason, a rock band travels in a tour bus instead of a small hatchback. Even though the hatchback is a bit faster through traffic; could fit 4 band members, the manager, and the equipment on the roof; and is considerably cheaper to build and operate, it's simply not a viable option for an extended journey.

A small hatchback is not made with long journeys in mind. It is made to be small, efficient, and a good car for short distances in urban environments.

Similarly, the supply ship is not built for an extended journey. It may be able to do its job by using a single battery, without any means of power generation, since it never stays in orbit for very long. If unmanned, it has zero accomodations for human life (space to live, air to breathe, ...). Maybe it doesn't even have a safe way of returning to Earth or even surviving reentry, because it's simply not worth the cost of developing those features.

Just because the supply ship could get there, doesn't mean that it's the best vehicle to use for the journey. It doesn't even necessarily mean that it's viable to do so.

Edit A small mention. Maybe you already figured this out, and this is the source of your question. Maybe not. I'll add it anyway, maybe for future readers.

If the supply ship had matched Hermes' velocity, delivered the goods, and then did not slow down or alter its course in any way; assuming Hermes also did not alter its course, the two ships would have travelled side by side and would have reached Mars at the same time.

So the core of your question, the supply ship being able to get to Mars just as quickly, is correct. You have simply forgotten to take into account the necessary machines and ship capabilities in order to sustain human survival for the entire time that the journey will take.

With regards to Mars, I believe Whatney hooks up with the ship before they slingshot back around the planet. For Whatney to intercept, their relative speeds just have to be close enough for him to be able to connect. It doesn't have to be a perfect match. They just have to be in close enough proximity as relatively similar speeds to intercept. Then speeds and trajectories can be slightly altered, or some sort of "grab" can happen, like they did with Whatney.

Going through the "slingshot," themselves, would not magically mean they matched the speeds or trajectories. The gravity is used to boost their velocity, but the effects would be different, depending on their course, their speed, their maneuvers while within the influence of the gravity, and the firing of engines before, during and after. It seems like you think about it kind of like dropping a can into a river, and letting the current carry you. It's not like that.

In any case, all the navigation is precisely calculated, so at any point and at any time, they should be able to anticipate both speed and trajectory, and calculate the resupply ship to intercept. This is no different than a ship or probe landing on a planet, an asteroid, docking with an orbiting station, etc. If I know exactly when, where and how fast the ship is going to be traveling, I can intercept it at the best location and time, and don't need to be following its same, exact footsteps to do so.

According to the source novel, Hermes decelerated for a month so that the Chinese booster could deliver the resupply probe:

“Iris is a point-thrust craft,” Venkat said. “Hermes has a constant-thrust ion engine. It’s always accelerating. Also, Hermes has a lot of velocity right now. On their current Earth-intercept course, they have to decelerate for the next month just to slow down to Earth’s speed.”
_{Chapter 16.}

And

“Well,” Venkat began, “if they did this ‘Rich Purnell Maneuver,’ they’d start accelerating right away, to preserve their velocity and gain even more. They wouldn’t intercept Earth at all, but would come close enough to use a gravity assist to adjust course. Around that time, they’d pick up a resupply probe with provisions for the extended trip.
_Ibid.

The Taiyang Shen is a rocket that was originally designed to bring a very heavy probe to a heliocentric orbit (i.e. to orbit the Sun, in this case between Mercury and Venus). Because of that , it has a lot of fuel (in technical terms: a lot of delta-V), to allow it to go all the way to Mars, or in this case - to intercept Hermes:

“It’s the Taiyang Shen’s booster, sir. Our engineers have run the numbers, and it has enough fuel for a Mars injection orbit. It could get there in four hundred and nineteen days.”
_Ibid.

And

“Earlier,” Teddy said, “you mentioned a supply probe for Hermes. We have that capability?”

“Yes, with the Taiyang Shen,” Venkat said. “We’d shoot for a near-Earth rendezvous. It’s a lot easier than getting a probe to Mars, that’s for sure.”
_Ibid.

And later for CNN's Mark Watney Report:

“Hermes isn’t going to orbit Earth,” Venkat said. “It’s just passing by on its way to Mars. And its velocity is huge. We need a booster capable of not only escaping Earth’s gravity but matching Hermes’s current velocity. Only the Taiyang Shen has enough power to do that.”
_{Chapter 19; italics author's, bold emphasis mine.}

As to how Watney was able to catch Hermes - it was by ditching a lot of weight, and by being very lucky. Firstly, the MAV had some fuel left from its initial landing. Watney made even more fuel, and ditched everything possible, even backups and some safety systems.

“Next up,” he continued, “We’re ditching all life support. The tanks, pumps, heaters, air lines, CO2 absorption system, even the insulation on the inner side of the hull. We don’t need it. We’ll have Watney wear his EVA suit for the whole trip.”

[. . .]

“We’ll remove the nose airlock, the windows, and Hull Panel Nineteen.”

Venkat blinked. “You’re taking the front of the ship off?”

“Sure,” Bruce said. “The nose airlock alone is four hundred kilograms. The windows are pretty damn heavy, too. And they’re connected by Hull Panel Nineteen, so may as well take that, too.”

[. . .]

“We’ll also have him remove the back panel of the pressure vessel. It’s the only other panel he can remove with the tools on hand. Also, we’re getting rid of the auxiliary fuel pump. Sad to see it go, but it weighs too much for its usefulness. And we’re nixing a Stage One engine.”

“An engine?”

“Yeah. The Stage One booster works fine if one engine goes out. It’ll save us a huge amount of weight. Only during the Stage One ascent, but still. Pretty good fuel savings.”
_{Chapter 25}

Notice that Hermes was also forced to remain on trajectory, and use the Martian slingshot. Otherwise, even with their ion engines, they'd take too long to return to Earth, and would die even with the supply probe present.

“It’s a flyby,” Venkat said. “Hermes isn’t going into Martian orbit. If they did, they wouldn’t be able to get back. They need their velocity for the return trajectory.”
_{Chapter 21.}