In a reputation based P2P network, an entity gains more resources or priority by having a higher reputation. Consider for example this site, Stack Exchange:
Suppose you could write a simple script that generates hundreds of new accounts in a minute, and generates proper questions for those accounts. You could have your bot accounts vote on each other's questions and answers and soon enough you'll be able to downvote anything to oblivion, control which questions are more visible, ...etc
As for Bittorrent, it doesn't implement any sort of shared reputation, but here's how it works:
if you generate enough identities, eventually, you should make it into everyone's DHT routing tables, and a bit later you'll have a sizable fraction of the total vote.
By replying with invalid info you can now pollute the network enough to introduce delays and outages. You could make it likely that anyone ends up with, say, 80% sybils, and now those sybils can give downloaders invalid file contents for certain torrents, effectively censoring them.
Source:
Liang Wang; Kangasharju, J., "Real-world sybil attacks in BitTorrent mainline DHT," Global Communications Conference (GLOBECOM), 2012 IEEE , vol., no., pp.826,832, 3-7 Dec. 2012