A "level set" of a function F is a set of the form {f: F(f)=a}, where a is a real number. So you can have the level set for a=0, and the level set for a=3, and the level set for a=-77, and so on.

In this particular case, since F(f)=f(0), your level sets are really {f: f(0)=a}. Is that an equivalence class of X/R? Is every equivalence class also a level set?