**Hint** : Write out $I_X(y)$ in terms of an integral using that $X$ has density $f_X$. Then apply Fubini's theorem.
* * *
**Edit** : The characeristic function $I_X$ can be written as $$ I_X(t)={\rm E}\left[e^{itX}\right]=\int_{-\infty}^\infty e^{itx}f_X(x)\,\mathrm dx $$ and plugging this into the left-hand side, we get $$ \int_{-\infty}^\infty I_X(y)f_Y(y)e^{-ity}\mathrm dy=\int_{-\infty}^\infty\int_{-\infty}^\infty e^{iyx}f_X(x)e^{-ity}f_Y(y) \,\mathrm dx\,\mathrm dy. $$
Now apply (the complex version of) Fubini's theorem.