Partial fraction decomposition of Laplace transform
Writing it out in full, for the general case, is tedious and I'm lazy, so just for illustration consider a simple form of \$H(s)\$, (and let \$jw_0 = jw\$ to save ink):

Let \$H(s) = 1/(s-p)\$, then \$Y(s) =  1/(s-p)(s-jw) = A/(s-p) + B/(s-jw)\$ for partial fraction expansion.

Solving for A and B: 

\$1 = A(s-jw) + B(s-p)\$

let \$s=jw\$, then \$B = 1/(jw-p)\$

let \$s=p\$, then \$A = 1/(p-jw)\$

Hence

$$Y(s) = \frac{A}{(s-p)} + \frac{1}{(jw-p)(s-jw)}$$

and we see that the coeff. of \$1/(s-jw)\$ is 

\$1/(jw-p) = H(jw)\$, therefore

$$Y(s) = \frac{A}{(s-p)} + \frac{H(jw)}{(s-jw)}$$