So input $$= R(s)$$ and output $$= C(s)$$ and forward transfer $$= G(s)$$ and feedback transfer $$= G_2(s)$$
deriving feedback now $$= G_2(s)C(s)$$ forward signal $$= R(s) - \text{feedback}$$$$ =R(s) - G_2(s)C(s)$$ output $$C(s) = G(s)(R(s) - G_2(s)C(s))$$
Now I don't understand this step The text says next is $$C(s) = \frac{G(s)}{1 + G_2(s)G(s)}R(s)$$
This is straightforward algebraic manipulation, which would be easier to follow if you just suppressed all those $\cdot(s)$ things. Anyway, I'll keep them for now.
Your penultimate equation
$$C(s) = G(s)\left(R(s)-G_2(s)C(s)\right)$$
can be re-written as
$$C(s) = G(s)R(s) - G(s)G_2(s)C(s)$$
(this is the distributive law), and moving the last term to the right yields
$$C(s)+C(s)G(s)G_2(s) = G(s)R(s)$$
or
$$C(s)(1+G(s)G_2(s)) = G(s)R(s)$$
and dividing through by $1+G(s)G_2(s)$ gives
$$C(s) = \frac{G(s)R(s)}{1+G_2(s)G(s)}$$
which is what you wanted.