With all the requests and assertions going back and forth, it can be cumbersome to debug issues with your SAML claims and assertions.
As within SAML a core principle is not needing a direct connection between the IdP and the SP, the user's browser acts as a message carrier between the two. Because of this all communication - albeit encrypted - goes through your own browser.
Using various debug tools you can see the exact communication and requests being made, and forwarded between IdP and SP.
To get you started, here are a couple of tools for various browsers that should get you started:
Using for example SAML Tracer you can see decoded SAML assertions and requests in real time while testing and debugging