Symmetry in general means that something stays the same in the face of some transformation. (This is a bit broad. Some symmetry is trivial and boring, and other is important. Anyone know a good statement which captures which is which better?)
Symmetry is a major concept in physics, in the form of conservation laws. For example, one says that total energy in a closed system stays the same when physical processes of all sorts take place.
Symmetry is also a major concept in philosophy. Philosophers watch out for symmetrical arguments or types of arguments (which argue equally well for the other side as the one they claim to support, or for all sides).
Symmetry also has an important role in aesthetics (in many cases, but not all, symmetrical things are more beautiful).
I think symmetry has a major role in math too, but I don't know the details.
Thus, symmetry is a concept with a lot of reach. It's an important concept.
What other fields is symmetry important to?
One reason symmetry may be important is that it's related to arbitrariness. Arbitrariness is the situation where all the choices look the same to us: they are symmetrical in every regard we know is important. And good explanations need to avoid being arbitrary. In general in philosophy, good explanations *break symmetry* in some way. But what does that have to do with physics, which has laws stating it's impossible to break certain symmetries? Or what does it have to do with aesthetics, where symmetry is often preferred?
Here's a bit more detail on symmetry. The best known kind is rotational symmetry. You take a picture and rotate it and get the same picture.
Conservation of momentum states that if have some total amount of momentum, and then you go forward in time (meanwhile doing whatever you want), then you have the same total momentum again. For example, consider two asteroids in deep space. Both are floating along. Then they collide, little pieces go flying everywhere. If you add up the momentum of all the pieces it's the same as the momentum from before the collision.
The general pattern is you have some thing, then you do some transformation process to change the thing in some significant way, but some key aspect stays the same.
With arguments, suppose you have an argument against X. Then you change it in some way, and now it's an argument against Y, but *everything else important stayed the same*. The symmetry is in the structure and logic of the argument, and the asymmetry is in its conclusion. For example, suppose I say it's bad to vote democrat b/c democrats are politicians and politicians are scumbags. This is a very bad argument because it's symmetrical with republicans or democrats as the targets. When we change it to "it's bad to vote republican b/c republicans are politicians and politicians are scumbags" the internal logic makes just as much sense before, it's exactly as compelling as an argument, only the conclusion has changed. So how can it support one of its possible conclusions over another equally valid one? It can't. So it fails.