How was this character able to catch Mjolnir and Stormbreaker?

Thanos never wields Mjolnir during Endgame.

What happens is Captain America takes a swing with Mjolnir at Thanos, Thanos grabs Captain America's hand (which is holding Mjolnir), and punches Captain America hard enough to knock him down. At this point, Mjolnir flies off to the side out of reach.

At no point does Thanos pick up/hold/carry/use Mjolnir in any way.

Thanos does not wield Mjolnir, and Stormbreaker has no such requirement

Thanos does not wield Mjolnir, he blocks it, deflects it, and battles against it but never picks it up and uses it. Therefore worthiness doesn't come into play here.

And Stormbreaker wasn't given any enchantments about worthiness, anyone who is strong enough to wield it can. Not even Mjolnir had any enchantments for worthiness until Thor's first movie when Odin placed it:

Whosoever holds this hammer, if he be worthy, shall possess the power of Thor.

_{Note: there is significant disagreement about whether Thanos actually wields Mjolnir and Stormbreaker. If he doesn't, no further explanation is required. For the purposes of this answer, I'll assume that he does and attempt to explain how this could be possible.}

The criteria for wielding Mjolnir and Stormbreaker aren't clear.

At the beginning of Thor (2011) Odin places this enchantment on Mjolnir:

Whosoever holds this hammer, if he be worthy, shall possess the power of Thor.

Unfortunately, Odin doesn't provide us with a handy guide of what "worthiness" means to him. Even after Thor becomes worthy at the end of Thor, he isn't exactly a paragon of good behavior. He's "heroic" in that he generally tries to do good, but he can also be arrogant¹ and irresponsible². The same descriptors apply to Tony, who can't wield the hammer. Neither can Rhodes, Barton, or Banner.

Option 1: Thanos is worthy

In an excellent answer to a related question, A J proposes that the worthiness comes from "being willing to sacrifice [oneself] to protect others." This isn't a airtight explanation³. Based on this criteria, Thanos might actually be worthy. At the beginning of Avengers: Endgame, he...

...destroys the stones because they are a temptation. Specifically, they are a temptation to the Avengers to destroy the new way of life that he has inflicted on the universe. The effort of doing this nearly kills him.

His logic is twisted, but his intentions are genuine. He believes he is protecting others from a grave threat, and he is willing to sacrifice himself to achieve this protection. It's possible that Mjolnir finds him worthy based on these criteria.

Option 2: Thanos is powerful

Thanos has been claimed to be "the most powerful being in the universe." It's not clear that this is literally true, but he is a remarkably powerful being able to do things that no one else can. We also know that Odin's magic isn't unbreakable: Hela is able to escape his imprisonment. This is speculation, but there's nothing logically inconsistent about Thanos being able to break or ignore Odin's spell.

¹ "Strongest. Avenger." (Thor, Thor: Ragnarok)

² "You were supposed to protect us. Asgard was supposed to protect us!" (Eitri, Avengers: Infinity War)

³ Just by being part of the Avengers, Barton and Rhodes risk their lives to help others. It's part of the job description. Futhermore, in Endgame, we see that...

...Barton is willing to sacrifice his life to get the Soul Stone.

Stormbreaker may not have the worthiness requirement, but I myself still think that Thanos was worthy. He genuinely believed that hat he was doing was the right thing to do. He was willing to sacrifice everything he loved (Gamora) and himself (the snap) in order to save half of the universe. He had a plan to save trillions, and he acted on it. He never wanted the stones for power, so after the job was done he destroyed them. These seem like actions that would prove oneself worthy of wielding the power of Thor.

There is a mythological nature to this question.

I am happy that some other persons have mentioned Hela, because as mentioned in various comments, Hela stopped Mjolnir in the air; Whether she wielded it or not is immaterial because the writers clearly intended not merely a statement about fandom canon, but about the nature of the hammer and Hela's character.

Namely, they used Mjolnir to demonstrate that Hela's nature and authority both predated and surpassed that of Mjolnir.

This was a statement not about Hela the character, but about the mythological character of death which she represents--that death itself both comes before the power of the worthy hero and is able to dominate him.

This was relevant to the plot because it showed a fundamental change in circumstance like those in many mythological works--Beowulf's battle with the dragon is a great example--this fundamental shift is perceived by the reader or viewer as a tragic but important statement about life itself which rings true to human experience.--The game changes, and even the greatest of mankind succumb to the tragic inevitabilities of human experience.

Likewise, the ability of Thanos to deflect Mjolnir is meant to be a similar indication by the creators; whether the authors' decision here was proper or not to the symbolic essence of Thanos is disputable--But the creators were trying to express that Thanos in himself possesses an authority which is in some way fundamentally primal and predates the authority symbolized by Mjolnir.

Yes, he is worthy, because looking at the science of Mjolnir, you have to be worthy to hold it in way, shape, or form. Remember how in Age of Ultrone, Tony couldn’t lift Mjolnir with his Iron Man hand.