Step 2 of proof by mathematical induction is to show that if a formula is true for any positive integer $n\geq n_{0}$, then it is also true for $(n+1)$. To proof the formula for “the sum of the first $n$ positive integers,” we add $(n+1)$ to both sides of the equation $1+2+3+...+n=\frac{n(n+1)}{2}$. We multiply the binomial expansion equation for $(a+x)^n$ by $(a+x)$ to show that it is also true for $(a+x)^{n+1}$.
Why don't we just substitute $n$ by $(n+1)$ instead? It's definitely easier that way. I know it's incorrect to do so, since we don't do that; but why is it an incorrect thing to do?
It's only valid to substitute $n+1$ for $n$ if you know that $p(n)$ being true implies that $p(n+1)$ is true; but proving $p(n) \Rightarrow p(n+1)$ is exactly what you need to do in the induction step!
You can certainly substitute $n+1$ for $n$ to figure out what it is you need to work towards in the induction step, but you can't just make the substitution and say you're finished.