= Solution
For the closed inclusion $f$, the <direct image sheaf> has stalk $\mathcal F_x$ at $x\in X$ and zero stalk at points outside $X$: outside the closed set there is an open neighbourhood disjoint from $X$. Hence $f_*$ is exact, since <exact sequences of sheaves> are detected on stalks.
Take a <flasque resolution> $\mathcal F\to\mathcal I^\bullet$. Direct image preserves flasqueness, because restrictions are the restrictions on inverse-image opens. Exactness makes $f_*\mathcal I^\bullet$ a flasque resolution of $f_*\mathcal F$. Moreover
$$
\Gamma(Y,f_*\mathcal I^\bullet)=\Gamma(X,\mathcal I^\bullet).
$$
The complexes computing <sheaf cohomology> are identical. Therefore
$$
\boxed{H^i(Y,f_*\mathcal F)\cong H^i(X,\mathcal F)\quad(i\geq0).}
$$
This is <sheaf cohomology under a closed inclusion>; the closedness hypothesis is used in the exactness of direct image.
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