I am trying to learn about the Adams Spectral Sequence and my question is basically summed up in the title.
More precisely, let $X$, $Y$, and $E$ be spectra. We have a homomorphism $[X,Y] \to Hom_{E^∗E}(E^∗Y,E^∗X)$, the latter giving the first page of the Adams Spectral Sequence. But why does $Ext^∗_{E^∗E}(E^∗Y,E^∗X)$ give a "better approximation" to $[X,Y]$ than $Hom_{E^∗E}(E^∗Y,E^∗X)$?
And more generally, it seems like magic that we can get homotopy from successively taking homology. Any intuition about this?