In 2 dimensions, the absolute value of a determinant of a matrix is just the area of the parallelogram subtended by the row vectors of the matrix. If this area is zero, it means that the vectors are co-linear. And of course the intersection of co-linear vectors is an infinite set (in R^n anyways). The same idea holds in higher dimensions, with the 3d case being similarly possible to visualise (as a volume). Since the unique solution of a linear system is the point at which a bunch of lines/planes/hyperplanes meet, a zero determinant implies a zero area which means that at least two vectors are coincident and hence there are infinitely many solutions.

[1] https://www.math.ucdavis.edu/~daddel/linear_algebra_appl/App...