Can you help me understand this pseudo random generator example?
Assuming your question is whether $G(x_1,x_2,...,x_n) = x_1,x_2,...,x_n, x_1 \oplus x_2 \oplus x_3 ... \oplus x_n$ 
is a PRG. 

$D$ is some fixed probabilistic polynomial time distinguisher.

So to prove that $G(.)$ is a PRG, you need so show that for all such distinguishers, $D(G(.))$ cannot be distinguished from $D(.)$. 

e.g.
$\big| Pr[D(G(s))=1] - Pr[D(s)=1] \big| \leqslant negl(n)$ where $s \leftarrow \{0,1\}^n $

Conversely to show $G(.)$ is not a PRG you simply need to identify a distinguisher $D(.)$ such that the above equation does not hold.

In this case, $G(.)$ looks like it could be distinguished from a random distribution, since the output is just the input, followed by the xor of the inputs.

i.e. $D(.)$ is some function which outputs $1$ if the final bit of output is the xor of all of the previous bits. This would be true half of the time applied to a random distribution, but would always be true for $G(.)$. This would mean that $\big| Pr[D(G(s))=1] - Pr[D(s)=1] \big| = \big| 1 - 0.5 \big| = 0.5$

Since a half is not negligible we can say that $G(.)$ is not a PRG.