Fleet PvP involves flying together with potentially hundreds of other players, to meet fleets of other players on the field of battle and destroy them.
Most large fleets are driven by battles between corporations. As such, taking place in a large fleet battle usually means being in one of the corporations involved.
Some of the better-known corporations which actively recruit new players and seek out large-scale fleet battles are Brave Newbies and Pandemic Horde. These battles can also be found in many, many corporations throughout New Eden; probably the best place to search is in sovereignty-holding nullsec alliances.
There are also some groups which accept pilots from any corporation into their fleets, and then set out to stir up trouble against any groups they come across (known as “NPSI” fleets, for “Not Purple, Shoot It”, referring to the fact that any non-fleetmate is a viable target) rather than trying to accomplish a strategic objective for their own benefit. The best-known of these is Spectre Fleet, but others certainly exist.
Alternately, the group Red Vs. Blue holds arranged fleet battle in high-security space, and focuses on new-player-friendly ships and relatively fair fights, offering the large fleet battle experience without the commitment of living in null-security space.
Large fleet battles are primarily driven by conflicts between player corporations and alliances. Conflicts can arise for a variety of reasons - desire to claim more territory and enjoy its resources, grudge matches over long-ago insults, mercenary contracts, or any of a countless number of others. By working together, you’ll have the chance to make a lasting mark on the face of EVE.