If I remember rightly it all depends which system is under attack or being attacked, ie if you have a battleship or not.
If you go to a contested system you can start fleet battle, either attacking the enemy fleet if your faction is defending, or destroying the enemy space station if your attacking yourself. If I remember there is an "attack starbase", or "attack fleet option on the mission screen when in a contested system.

if you have a battleship or carrier class you can start attacks yourself, anything smaller like a cruser and you'll just have to go to the right system at the right time.

Eventually you can take over the enemy capital and win the war which is cool! .

One other thing to bare in mind with fleet battles, is that usually your escort ships are pretty crappy, so unless you select one specifically to attack a ship of the same class on the other side, your escort will probably lose. Also, don't ever select an escort to fight an enemy missile or you can say bye bye to the ship in one hit.

Hth.

Sorry if I'm misremembering, while I played s5 and s5 invasion a hell of a lot back in 2014 or so I've not played for quite a while since.