The equation has a typo on the right hand side - it should be:
$\displaystyle \prod\limits _{r=1}^n (1+x^{2^r}) = \frac {1-x^{2^{n+1}}}{1-x^2} $
(note that the exponent in the numerator on the right hand side has 2^(n+1) in it, not 2^(r+1).
Using induction you first prove that this is true for n=1. Then show that if it's true for n=k it's also true for n = k+1.
Given: $\displaystyle \prod\limits _{r=1}^n (1+x^{2^r}) = \frac {1-x^{2^{n+1}}}{1-x^2} $, then
$\displaystyle \prod\limits _{r=1}^{k+1} (1+x^{2^r}) = \left( \prod\limits _{r=1} ^{k} (1+x^{2^r}) \right) (1+x^{2^{k+1}}) = \frac {1-x^{2^{k+1}}}{1-x^2} (1+x^{2^{k+1}}) $
which equals $\displaystyle \frac {1-x^{2^{k+2}}}{1-x^2}$