Can someone please explain why we are allowed to substitute $b=2f(a)$? the statement is underlined in red. What about this situation allows one to make this substitution? Like I cant say, let $af(a)+bf(b)=2f(a)f(b)+1$ because then the inequality obviously wouldn't hold. This substitution doesn't make any sense to me.
You have shown that inequality $2f(a)f(b) \ge af(a) + bf(b)$ holds for any choice of real numbers $a$ and $b$. Since you are free to let $b$ be whatever you like, you can choose $b=f(a)$.