Did you copy that correctly?
If it's supposed to be
This was already shown
So,
This gives us
Sorry to have to say so, but the expression in red makes no sense (nor does the rest of that argument). if k is the summation index, then it cannot also occur as part of one of the limits of summation. The summation index is a variable, and the limits of summation must be constants.
The left side of that identity is the coefficient of in the expression . But that is a geometric series, with sum . To find the coefficient of there, write .
Picking out the coefficient of in , you find that it is
But in the second line of that expression, the sum of all the binomial coefficients is . In the first line, the sum of half of the binomial coefficients (with k going from 0 to n) is . That gives the value for the coefficient of as