If you know what a Young diagram is and how it relates to partitions, here's a neat argument:
Take a $n$ by $n$ box and put a Young diagram partitioning n on the right side, and one underneath it. What do you get? a Young diagram partitioning $n^2 + 2n$. What if you take the box away? You get a pair of Young diagrams each partitioning $n$. This gives an injection of pairs of $n$ partitions into partitions of $n^2 + 2n$
If you don't know about Young diagrams ( you should learn them! They're really interesting, fun, and important ), the same argument can be made less geometrically. Take any two partitions of $n$ and add $n + n + n + \ldots + n$ ( $n$ added $n$ times ) to the first partition. What you get is a partition of $n^2 + n$ with parts all greater than or equal to $n$, so you just add your other partition of $n$ to that, and you get a partition of $n^2 + 2n$. It's not too hard to see this algorithm is injective.
Example where $n = 6$
$$ f = 3 + 2 + 1 $$
$$ g = 3 + 3 $$
$$ h = 9 + 8 + 7 + 6 + 6 + 6 + 3 + 3 $$
So $f$ and $g$ are partitions of $n$ and $h$ is a partition of $n^2 + 2n$ formed in the way described above.
Really the Young diagram version is cuter though!
Oh, to get strict inequality, just take the partition which is all ones of $n^2 + 2n$