There are two important things to know about the FAFSA® / federal financial aid and studying abroad:
About 750 international institutions are eligible for US federal financial aid. You cannot receive or use US federal financial aid for any school outside this ~750. You can find the list of ~600 European colleges here.
Assuming you are attending one of the international colleges eligible for aid, you are only eligible for federal loans (Direct Subsidized, Direct Unsubsidized, and Parent PLUS). You are not eligible for any federal government grants (Pell Grant, FSEOG). You will also not be eligible for state aid/grants, since those are reserved almost exclusively for students attending college in-state.
If you would like to be considered for those federal loans (remember: the Direct Loans often have much better rates than what you'd find on the private market), you'll need to fill out the FAFSA®.
Also, if you're applying to a combination of international colleges and US colleges, you'll need to fill out the FAFSA® just in case you end up going to a college here in America.