Why are the heroes of Overwatch fighting?

From an interview with Michael Chu, a Senior Game Designer on Overwatch (emphasis mine):

If you look at a game like World Of Warcraft, the Horde/Alliance split is such a big deal and such a defining aspect of the game. But Overwatch, pretty much from the start, we didn’t want to restrict ourselves that way. Because the game is so much about having the flexibility to play all these heroes and finding all the ways they play well together or against each other. Because of that we decided that the gameplay of the Overwatch universe wouldn’t be bound by that.

Our story justification – what I usually tell people – is that I think of it as, it’s clearly not canonical, but I think of it as how in comic book stories sometimes Iron Man and Captain America fight each other. Maybe in this instance on this map they’re fighting – for some reason they have some ideological difference or something that’s bringing them against each other.

But I think ultimately it was important for the game that you were able to play whatever characters you want. Blizzard has this development philosophy which is ‘Gameplay first’ and this is definitely an example of where we’ve gone with that.

Overwatch's story comes in the form of the media surrounding the game, including short animations and comics. Prior to the release of the game, Blizzard Watch was keeping up to date with all the individual media pieces and putting the story together, such as this great article about the origins of Reaper.

To give a short answer: They are fighting because it's fun for us outside of the game's narrative. The characters in the game do fight within the story narrative, for many reasons, but it is triggered with the creation of Omnics, robots created to serve humanity. Eventually the Omnics rebel and Overwatch helps to bring them back down again. Certain team members are rewarded with promotions for their performace in the war, which does not go down well with others in the team. (I'll let you read about the rest for yourself on Blizzard Watch.)

A side note: To add the story of Overwatch in to the game would make no sense logistically, at least not in the games current state.

Some of the characters in this game are bad, and I mean bad (go take a look at Widowmaker's origins.) And then we have Tracer and Winston and Soldier 76: staples of the good in the world. And then there's some in-between, some who do or do not agree with the Omnic-Human harmony but cannot decide how to take a stance. (A number of character combinations activates dialog between characters at the start of the game. Some of these characters make derogatory comments to other Omnic characters in the game.)

But outside of the games narrative, we need to be able to use a mixture of these good and bad characters to build teams together. To segregate the characters in to teams would ruin potential builds and effect game play.

The way Blizzard is handling the story is awesome. At the end of the day there is a narrative, and it is being given to us within beautiful videos and brilliant comics.

There is a story but that is completely ignored in the game there isn't any explanation to why they are fighting.

Winston recalls Overwatch because he feels the world is in trouble and it needs them. The "Are You With Us" video shows quick cuts of several violent incidents, including buildings on fire and under attack. It is not clear what is causing the violence.

source: Overwatch Cinematic Teaser | "Are You With Us?"

Apparently, each map does have an objective in the story, from what I've heard, but I have no idea where people found them.

• On King's Row, attackers are anti-Omnic terrorists that must escort an EMP bomb to kill all Omnics in London.
• On Dorado, IIRC, the attackers are delivering weapons to the gang seen in the Hero short.
• On Numbani, the attackers are escorting the Doomfist Gauntlet back to the museum. Defenders are looking to steal it.
• On Hollywood, the attackers are escorting an Omnic movie director (Hal-fred Glitchbot, the same guy who has his name on some of the posters in the attacker spawn) to the filming scene of his next movie. The defenders are anti-Omnics. Yes, it's stupid.

That's the only ones I know.

As of why they're fighting, shit's going off the hook like shit and the hook were having a violent divorce due to tensions between Omnics and humans and rising crime everywhere. Overwatch's activity has been made illegal years ago, but that's not gonna stop them from doing something to make the world a less awful place.