Why Superman and Batman fight?
It's something of a comic book tradition that, before two major heroes team up, they inevitably end up fighting each other. It's a bit of fan service, watching two super-powerful characters go at it, but it's also a way of establishing mutual respect in the mind of the other hero. "Ok, ok, so obviously you can handle yourself. I guess we can work together."
A famous example of this was the first Avengers movie in the MCU, in which Thor faces off in a (fairly brutal) exchange against Cap and Iron Man, before they grudgingly allow him to join their team.
Given that Batman and Superman are the two biggest heroes in the DC Universe, it seems that the DCCU will be devoting an entire movie to this "opening phase" of their relationship. The newly-released trailers make it CLEAR that this is a real conflict, with them facing off against each other in combat for at least a significant part of the film. Presumably they'll become allies by the end, but they do seem to be enemies at first.
Remember, when Superman met everyone in the New 52, they ALL ended up fighting each other:
So basically, it's just what happens when you get a bunch of Type-A personalities who fight people for a living, and put them in a room together.
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