"Si" is here because the verb is reflexive: esercitarsi = to train oneself http://italian.about.com/library/fare/blfare118a.htm

So why not "lei esercita di notte" ?

Esercitare is a transitive verb (a verb that requires one or more objects). So taking that in regard and the fact that we don't have an object in our sentence we can safely rule out esercitare

I'm not 100% sure in my answer so I hope someone with better knowledge of Italian grammar helps us :)

Yes, kind of.

I am not a native English speaker but it sounds a bit weird to me. That "herself" is redundant.

If you wanted to say "She exercises by herself at night" that would be "Lei si esercita da sola di notte".

Literally, yes. This is also true in, e.g., "si è persa" ("she [herself] is lost"). Like dnovinc mentioned, though, that's probably not how you'd want to translate it.