I just don't understand how this equation breaks down like this. The second step... $[k^2 + k + 2k + 1]$ is perplexing, but breaking that down to $(k+1)(k+2)$ has completely baffled me. I would expect this to be equivalent to $[k^2 + 2k + 1k + 2]$.
Any help would be greatly appreciated.
There is a typo in what you showed -- it should be $k^2+k+2k+2$, not $+1$. So indeed, you're right to be baffled.
After fixing this, here what the derivation does: you have $$ \frac{k(k+1)}{2} + (k+1) = \frac{k^2+k}{2} + \frac{2k+2}{2} = \frac{k^2+3k+2}{2} $$ explaining the first step.
Now, you have that $$(k+1)(k+2) = k(k+1)+(k+2) = (k^2+k)+(k+2) = k^2+3k+2$$ explaining the second.
You could have directly factored $5(k+1)$ for a shorter computation: $$\frac{5k(k+1)}2+5(k+1)=5(k+1)\Bigl(\frac12k+1\Bigr))=\frac52(k+1)(k+2).$$
