From the definition as lines you have that a point $(X,Y,Z)\neq(0,0,0)$ determines a line through the origin.

If the point happens to have $Z\neq0$ then the point $(X/Z,Y/Z,1)$ determines the same line through the origin.

This means that $(X,Y,Z)$ and $(x,y,1)$, with $x=X/Z, y=Y/Z$, are representing the same point of the projective plane.

If we take $X^2+Y^2=Z^2$ and look only in the part of the projective plane where $Z\neq0$ then we can divide both side by $Z$ and get $(X/Z)^2+(Y/Z)^2=1$, which is the equation $x^2+y^2=1$ in the coordinates $(x,y)$.

Of course, the points such that $x^2+y^2=1$ are only those points for which $Z\neq0$ (the complements of a line in the projective plane).

To have the complete picture we also need to study it at points of the projective plane where $Z=0$. So the solutions of $x^2+y^2=1$ are not the whole picture.

If $Z=0$,then either $X\neq0$ or $Y\neq$. Since the equation is symmetric with respect to $X,Y$, it is enough to look at the case $Y\neq0$. In this case we can proceed similarly. We divide by $Y^2$ and get $(X/Y)^2+1=(Z/Y)^2$. Calling $\tilde{x}:=X/Y$ and $\tilde{z}:=Z/Y$ we get $\tilde{x}^2+1=\tilde{z}^2$.