The general approach for proving that something is not a PRF is to come up with a distinguisher for it. A general approach for proving that a construction $F'$ from a PRF $F$ is also pseudo-random is to prove the contrapositive. That is we prove the contrapositive of
$F$ is pseudo-random implies $F'$ is pseudo-random
$F'$ is not pseudo-random implies $F$ is not pseudo-random.
This is easier to prove as we assume that there is some distinguisher $D'$ which distinguishes $F'$ from a random oracle, and then prove that this implies there is some distinguisher $D$, distinguishing $F$ from a random oracle. To do this, we can show that if $D$ is playing the PRF game against a challenger who has chosen $F$ or a random oracle, then $D$ may simulate (in a polynomial number of queries to the challenger) the PRF game between $D'$ and a challenger who has chosen $F'$ or a random oracle. You could think of this interaction as though $D$ has $D'$ in it's belly, and uses it to win the PRF game as shown in the picture
To get an intuitive sens of how this "reduction" would work for this example, imagine that for each query $x_i$ that $D'$ makes, $D$ takes that query and queries the challenger twice (once for $0||x_i$, and once for $1||x$) returning, the concatenation of both of these queries to $D'$. This has effectively made it so that $D$ has returned $F'_k(x_i)$ to $D'$, and in this way simulates that $D'$ is interacting with a challenger for the PRF $F'_k$. Since this simulation was done with a polynomial number of queries (2x as many), if $D$ responds with the same answer that $D'$ produces, it will have the same advantage as $D'$. Thus, if $D'$ has non-negligible advantage, so does, $D$, which is the contrapositive we sought to prove.