I was reading notes on the Fundamental Group and came across the statement $Z*Z/({xyx^{-1}y}) = 1$. Why is this true?
If a quotient group is trivial doesn't it mean that the two factors are isomorphic meaning $Z*Z \simeq {xyx^{-1}y}$
Link to the notes: http://math.arizona.edu/~glickenstein/math534_1011/vankampen.pdf
You're misreading the (admittedly rather poor) notation. The statement isn't that $\Bbb Z * \Bbb Z / \langle xyx^{-1} y\rangle$ is the trivial group. (The $xyx^{-1}y = 1$ bit is supposed to invoke the idea that we're setting $xyx^{-1}y$ to one and seeing what we get.)
The author is saying that the fundamental group of the Klein bottle has presentation $\langle x, y \mid xyx^{-1}y \rangle$, which means that it's the quotient of the free group $F_2$ on two generators $x,y$ by the normal subgroup 'normally generated' by $xyx^{-1}y$; that is, the smallest normal subgroup of $F_2$ containing this word $xyx^{-1}y$.