Many of the references that people have mentioned are very nice, but the brutal truth
is that you have to work **very hard** through some basic examples before it really makes
sense.

Take a complex $K=K^\bullet$ with a two step filtration $F^1\subset F^0=K$, the spectral
sequence contains no more information than is contained in the long exact sequence associated
to
$$0 \to F^1\to F^0\to (F^1/F^0)\to 0$$
Now consider a three step filtration $F^2\subset F^1\subset F^0=K$, write down all the short
exact sequences you can and see what you get. The game is to somehow relate $H^*(K)$
to $H^*(F^i/F^{i+1})$. Suppose you know these are zero, is $H^*(K)=0$? Once you've mastered
that then ...