How can a $C^1$-continuous surface have infinite curvature?

Short answer: the curvature is a property of the second derviative, and it's perfectly possible for a $C^1$ function to have a badly behaved second derivative.

In one variable, think about the function $f(t) = |t|^{3/2}$; it is $C^1$ but $f''(0) = +\infty$.

To turn this into a surface with infinite curvature, try $$X(u,v) = (u, |u|^{3/2}, v)$$ i.e. just extending this curve in the $z$ direction, forming a sheet. You'll find the normal curvature at the origin in the direction normal to the $xy$-plane is infinite.