How can Spider-Man watch himself fighting with Ant-Man in the trailer of Spider-Man Homecoming?

Any number of reasons.

1. Iron Man and War Machine. The suits have cameras as seen in IM3 remote control flight.

2. Spider suit has mechanical lenses. Likely a camera, thanks to Stark Tech.

3. Airport security cameras.

4. Any number of spy satellites monitoring the fight, including Stark satellites.

5. Parker set up cameras. <-- once the movie released, we found the answer.

6. Hank Pym had spy ants set up.

7. Fury had better cameras around.

8. Falcon has a camera in his headset. And in Red Wing, his drone.

9. Black Panther suit likely has cameras.

10. I'd be shocked if Vision can't record or upload.

11. Suspension of disbelief and the typical use of non-realistic video footage of previous movies. The in universe explanation is hand waved.

Airports do have copious amounts of security cameras, as DForck42 suggests.

Also, Peter Parker (in previous movies and the comics) has an established habit of setting up (photographic) cameras to capture pictures of himself as Spiderman that he can sell to his newspaper. In fact, in some Spiderman continuities, this is how he first earns his position as a freelance photographer, and thus makes a living for himself. So, it's not outside the realm of suspension of belief that he did actually set up a video camera or two to film himself fighting in Civil War. Strung up in otherwise improbable positions with webbing, naturally.

It was revealed in Spider-man Homecoming's first scene of Peter:

Peter himself shown obsession with vloging and recording it himself, he must have edited it and uploaded it on YouTube.

_{Note: But Peter was unaware that his suit had ability to record everything too which he got to know about later.}

Its revealed in the intro of Spider-Man Homecoming. He's making a vlog.