Fleet actions are large-scale PvE battle areas. They're generally big open maps that can accomodate something like 20-40 people, possibly more.
When you enter one, there's a set of mission objectives that everyone in the area can view - and as they're completed, the mission objectives update for everyone. Regardless of whether you're teamed.
When the Fleet Action is complete, there's a ranking of top performers - this is slightly skewed, because you place higher in the rankings based on damage dealt (therefore it rewards DPS players over healers/buffers).
But the end of the Fleet Action will give you a reward item, and if you place high in rankings, you should get better stuff.
I actually made my own fleet that caters to my own tastes so I doubt I will get much people in it, hence why I was worried fleets were required for anything big.
In City of Heroes, I have my own supergroup for all my characters and team with acquaintances and random groups while in Villains, I do have a supergroup and like it but I do like having control over my own base, etc. from time to time.