Use "claim to ___" when you want to tell what a person said, but you also want to suggest that you don't believe them, or don't know if what they said is true. For example:

I once knew a guy who claimed to be able to remember every telephone number he'd ever called.

There are two different patterns that people use "claim" in. One is "claim that ___", and the other is "claim to ___":

They claimed to be police officers, so I let them in.

They claimed that they were police officers, so I let them in.