This has a beautiful geometrical interpretation. Note $\rm\: x,y\: $ is a $\rm\:\mathbb Z$-basis of $\rm\: \mathbb Z^2\, $ iff $\rm\: \mathbb Z^2\: $ is tiled by the fundamental parallelogram $\rm P $ with sides $\rm\,x,y.\, $ But this is true iff the only lattice points that are inside $\rm P $ or on the boundary of $\rm\:P\:$ are its vertices. However, by Pick's area formula, this is true iff $\rm\ area\ P =\: $ #interior_points $\, +\,\ 1/2\, $ #boundary_points $-\ 1\! =$ $\, 0 + 4/2 - 1 = 1.\,$ But by basic analytic geometry $\rm\: area\ P\: =\: |\det(x,y)|.\,$ Therefore,combining the two,we conclude that $\rm\: x,y\:$ is a $\rm\:\mathbb Z$-basis of $\rm\: \mathbb Z^2\! \iff |\det(x,y)| = 1\:.$
In fact it deserves to be much better known that Pick originally applied his area formula in a similar way to give a beautiful geometric proof of the Bezout linear representation of the gcd.