Well, it depends on what you mean by "easy". A special case, which I find very instructive, is a theorem of Deligne from the late 1960's. 

> Theorem. $\mathbb{R} f_*\mathbb{Q}\cong \bigoplus_i R^if_*\mathbb{Q}[-i]$, when $f:X\to Y$  is a smooth projective morphism of varieties over $\mathbb{C}$. (This holds more generally with $\mathbb{Q}_\ell$-coefficients.)

> Corollary. The Leray spectral sequence degenerates.

 The result was deduced from the hard Lefschetz theorem. 
An outline of a proof (of the corollary) can be found in Griffiths and Harris.
It is tricky but essentially elementary. 

A much  less elementary, but more conceptual argument, uses 
weights. Say $Y$ is smooth and projective, then 
$E_2^{pq}=H^p(Y, R^qf_*\mathbb{Q})$ *should* be pure of weight $p+q$ (in the sense of Hodge
theory or $\ell$-adic cohomology). Since 
$$d_2: E_2^{pq}\to E_2^{p+2,q-1}$$
maps a structure of one weight to another it must vanish. Similarly for higher differentials. 

If $f$ is proper but not smooth, the decomposition theorem shows that $\mathbb{R} f_*\mathbb{Q}$ decomposes into sum of translates of intersection cohomology complexes. 
This follows from more sophisticated purity arguments (either in the $\ell$-adic setting as in BBD, or the Hodge theoretic setting in Saito's work).
There is also a newer proof due to de Cataldo and Migliorini which seems a bit more geometric. 

I have been working through some of this stuff slowly. So I may have more to say in a few months time. Rather than updating this post, it may be more efficient for the people
interested to check 
<a href="http://www.math.purdue.edu/~dvb/seminar.html"> here </a> periodically.