Maybe it gives you the opportunity to network with higher-profile contributors to your field, if you offer to peer review their work (though to be honest, I don't know how the peer review process works, and if you can even "offer" to perform it for a specific paper).
Peer review is not the same thing as replication, which is replicating the results of a paper after it's been published (e.g., confirming some groundbreaking finding). Peer review happens at the stage before publication of the original paper. Researcher(s) submit the paper to the journal. The journal editor sends the paper out to some reviewers, who review the paper (this is the "peer review" stage). Pending reviewer feedback and editor approval, the paper is published.
Edit: also the "benefit" that rsa4046 refers to probably doesn't mean networking. AFAIK, reviewers are always anonymous to the authors (which can generate its own problems e.g., if the reviewer gets a paper authored by someone he/she doesn't get along with). The benefit being referred to, I believe, is that of learning to write better reviews, and having reviewed other's work, learning how to improve your own.
The authors most likely won’t (can’t) ever know that I agreed to review their paper; it’s more of a good citizen affair. Sometimes, afterwards, it will become apparent that a particular referee was someone familiar. I would like to think that the original poster was imagining something along those lines, but your take is probably closer to the truth. Oh well.