= Solution
The map $H\times K\to HK$, $(h,k)\mapsto hk$, has fibers of size $|H\cap K|$: two pairs have the same product exactly when their quotient is represented by an element of the intersection. Hence
$$
|HK|=\frac{|H||K|}{|H\cap K|}.
$$
The standard intersection-index inequality gives $[G:H\cap K]\leq [G:H][G:K]=ab$. This index is divisible by both coprime numbers $a$ and $b$, so it equals $ab$. The displayed formula then gives $|HK|=|G|$, hence $HK=G$.
The conjugacy class and centralizer are
$$
\operatorname{Conj}_G(x)=\{g^{-1}xg:g\in G\},
\qquad C_G(x)=\{g:gx=xg\},
$$
and orbit--stabilizer gives $|\operatorname{Conj}_G(x)|=[G:C_G(x)]$. Thus the coprimality hypothesis gives $C_G(x)C_G(y)=G$.
One inclusion in the desired equality follows by conjugating $xy$. Conversely take $g^{-1}xg\,h^{-1}yh$. Write $hg^{-1}=ab$ with $a\in C_G(y)$ and $b\in C_G(x)$. The hint then rewrites the product as
$$
h^{-1}a(xy)a^{-1}h,
$$
which is conjugate to $xy$. Therefore
$$
\boxed{\operatorname{Conj}_G(xy)=
\operatorname{Conj}_G(x)\operatorname{Conj}_G(y).}
$$
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