Derive DFT of $x[2n]$
You haven't specified what part exactly of the solution you didn't understand.

The purpose of the posed question is to show the spectral effects of signal downsampling, in this case by a factor of 2.

The original discrete signal is $x[n]$ (whose DTFT is $X(e^{j\omega})$), with a length of N samples, and the downsampled signal (by a factor of 2) is $g_7[n]=x[2n]$, so $g_7[n]$ is made from the even samples of $x[n]$, with a total length of N/2 samples.

The first experssion of the solution is simply the DFT (by definition) of $g_7[n]$. The solution then proceeds to some series manipulation, then uses the DTFT-derived definition of DFT to show the final expression.

The final expression of the solution shows that downsampling the signal by a factor of 2 produces a double-frequency image of the original signal in the frequency domain. This alludes to the importance of anti-aliasing filtering prior to signal downsampling.