An idea of the proof. For he convexity of $f$: a function is convex iff its epigraph is convex; the epigraph of $F$ is the projection of the epigraph of $f$; the projection of a convex set is convex. Note that this part also work with $\inf$ more generally that $\min$ in the definition. 

For the strict convexity: any minimum point of $f$ is the projection of a minimum point of $F$, so if $f$ has more than a minimum point, so does $F$, and $F$ is not strictly convex. Up to adding  a linear form to $f$, the latter is the case when $f$ is not strictly convex.

Finally, note that a strictly convex and bounded below function $f$ does not produce in general a strictly convex $F(x):=\inf_y f(x,y)$, like the example of $f(x,y):=\exp(x^2/2+y)$ shows, since $\operatorname{det}D^2f=f^2>0$ and $F$ is identically $0$.