With the inductive hypothesis, we assumed that n^{3}+2n was divisible by 3 for n, and now you're proving the same for n + 1.
It's like knocking down a line of dominoes, if we can prove that the first domino falls over (basis) and each domino knocks over the next (inductive step), then that means that all of the dominoes in the line get knocked down eventually.
You know that (n^{3}+2n)+3(n^{2}+n+1) is divisible by 3 because n^{3}+2n is (because of the inductive hypothesis) and 3(n^{2}+n+1) is (because it's 3 times an integer). So, their sum is as well.
