How did NASA figure out Mark was still alive?

The reason they don't look at the Ares 3 site is that they don't want to get an image of Mark Watney's body which they would have to release to the public (since NASA is a government organisation it has to release all its images to the public).

After two months they finally take a look because they hope they can maybe start an additional mission which can use the resources left (and bring back the body).

And then it takes just one picture to see that the rovers have been moved and the solar cells have been cleaned.

(In the book they also see that some emergency tents were attached to the hab (set up to get more space to grow potatoes) and the remains from the descent and ascent vehicles have been salvaged. But this is not mentioned in the film.)

In the film, there's a scene between Kapoor and Sanders, immediately before Park receives the message from Kapoor, which explains the situation (covering much of the novel information from Raidri's answer):

1. The Ares 3 mission was scrubbed a little past half of the way into its projected 30-sol time frame due to the storm, so it was largely a waste.

2. Mars missions are massively expensive, and NASA has to beg for every cent of funding. If there is any salvageable equipment at the Ares 3 site, then there's that much less that the Ares 4 or 5 astronauts would have to bring with them; this would greatly reduce overall mission costs. The HAB was sent with 68 sols of supplies, typical NASA redundancy.

Because of those two points, Kapoor states that they need to check immediately to see what equipment might be left by taking satellite photos of the area. Sanders states that these photos should not be taken because they will be available to the public, and may show Watney's corpse. This gives Kapoor another point for his argument:

3. If the next mission includes a plan to retrieve Watney's corpse, then they'll have a much easier time obtaining funding; there would be a public outcry at the thought of abandoning the body there, only if the public hears about it sooner than later. As Vincent put it, in a year, no one would give a S-word.

So it all circled back around to "we need photographs of the site before we can plan anything or beg for any funding". Sanders granted permission to Kapoor with the assumption that Watney was dead, and Kapoor passed his request to Park with the same assumption.

Answer according to the film:

Raidri has it spot on for the book version.

The film version, likely due to time constraints, had to gloss over a lot of these details that were in the book.

In the film version, they discover that he is still alive by noticing two things on the satellite imagery:

• The Rover has been moved from where it was parked originally (prior to the evacuation)
• The solar panels had been cleaned.

Raidri's answer points out that in the book, they also spotted tents attached to the hab for growing potatoes. In the film, the potatoes were grown completely inside the HAB, so no external tents would have been seen by the satellite.

The Satellite Time reason given by the NASA Director to Vincent was a PR excuse, a way to try and save face. It would allow plausible deniability and prevent NASA, the Ares 3 crew and himself from dealing with any negative press from the actions on Sol-18 of the mission. As he explained to Vincent when pushed in the same scene, NASA is a Public Domain organization, they have to by law release any copyrightable image into the Public Domain, and:

Congress won't reimburse us for a paperclip if I put a dead astronaut on the front page of the Washington Post [newspaper].

The Director wanted to wait a year for normal weather activity to bury Mark in sand before authorizing any Satellite time.

Vincent immediately frames the issue in terms that the Director understood, and ultimately agreed with. Get him the pictures of Mark's body now, and he can sell it to the public and Congress while the public still cares about it. In a year, no one would give a shit. He dangled a free check in the Director's face and the Director lunged at it, giving Vincent the Satellite time he requested.