Why does the natural ring homomorphism induce a surjective group homomorphism of units?

First reduce to the case where $n$ and $m$ are both powers of a prime $p$, using the Chinese remainder theorem. After this reduction the result is clear since the $p$-adic valuation of $n$ is lesser than the $p$-adic valuation of $m$ for any prime $p$.

If you want a "pedestrian" solution without knowledge in group theory or number theory, you have to note that when $k$ is your unit, then $k+ln$ are also units, so you have several candidates for the unit in the bigger group and you only have to plug them into your equation and show that one of them gives a coefficient of $n$ divisible by $a$.