For example, to write music and use it as the soundtrack to a monetized YouTube video. Or produce a song that you sell on a streaming service like iTunes.

There is an issue with YouTube where even if you make music with phrase-based software you own (Sonokinetic, The Orchestra, etc) or loops, you can still be caught by Content ID from somebody else who made music with the same tools and registered it. But that's a whole different issue.

Does buying BIAB give you the rights to use in products that you are paid for?

Thanks!

Yes.

Welcome Tiger The Frog! Anything you create with Band-in-a-Box is considered your song, unless of course if you use it to create a cover song. If you'd like this info in writing, we had a post about copyright in our Sales FAQ here: https://www.pgmusic.com/salesfaq5.htm#23