How to understand RSA Proofs of correctness at Wikipedia
If $m^{ed} \equiv m \pmod{q}$, then $q\;|\;(m^{ed}-m)$;
if $m^{ed} \equiv m \pmod{p}$, then $p\;|\;(m^{ed}-m)$;
thus $m^{ed}-m$ is a multiple of both $p$ and $q$;
thus $m^{ed}-m$ is a multiple of $\operatorname{lcm}(p,q)$.

Because both $p$ and $q$ are distinct primes, $\operatorname{lcm}(p,q)=pq$ holds;
Thus, $m^{ed}-m$ is a multiple of $pq$.

So $(pq)\;|\;(m^{ed} - m)$ clearly holds.
Thus finally, $m^{ed} \equiv m \pmod{pq}$.