Fraud proof is simply some compact data telling you that Bitcoin rules were violated. For many of the rules we can create compact proofs that they were violated.
Nope. Still not getting it. A block with all associated signatures already contains everything needed to determine whether or not it is fraudulent.
It does, but not everything can be proven fraudulent compactly at the moment, in a way compatible with lightweight clients. Fraud proofs are allowing for a lite client to be aware of bad blocks without fully downloading and verifying them.
By fully verifying the blockchain you are proving that it's valid (good). By using fraud proof you are proving that something is
invalid (bad). Notice the difference.