You must be already tracking related pages, competitors on Facebook with tools like Recommend.ly. Interacting on posts made on related pages brings visibility to your Page with fans and fellow admins. Make sure you pick only relevant conversations and try to add value to them. This helps you increase traffic and fan count.

HOWEVER, avoid the temptation to spam high-traffic Pages in the name of interaction. You will just dilute your brand value for temporary gains in fan count. Make sure the whole interaction is relevant to both you and the other page.