The first direction is easy. Determinant is a continuous function from the matrices into the reals. If a continuous path carries V to W, and their determinants have opposite signs, the determinant of an intermediate matrix is 0, and that matrix is not a basis. Therefore the two determinants must have the same sign.
Consider the reflection of the standard basis that maps x to -x. This changes the determinant from 1 to -1. We must squash all of n space down into the hyperplane at x = 0, then back out the other side, like pushing something through a mirror. For a brief instant in time, space is compressed into a plane, in the flat mirror so to speak, and at that time our basis isn't a basis. It doesn't span the entire space.
Assume det(V) and det(W) have the same sign. Use gaussian elimination and back substitution to carry V onto a diagonal matrix, then down to the identity matrix. Do the same for W, then reverse it and connect the two paths together. This takes us from V to the identity matrix to W. Let's make sure we can do this continuously.
If we must subtract twice the fourth row from the fifth, do it gradually. Subtract 1% of the fourth row, then 2%, and so on, until we reach 2.0. This is a continuous function on V, and it leaves the determinant unchanged. thus V is a basis at all times.
Rows can also be swapped continuously. To swap the first and second rows, subtract the second from the first, add the first to the second, and subtract the second from the first. This leaves the determinant unchanged and negates the first row. We use to simply swap rows and remember the -1; this way we don't have to remember the -1. It is incorporated into one of the swapped rows.
After back substitution we have a diagonal matrix. Gradualy scale each diagonal entry down to 1 or -1. This changes the determinant of course, but it preserves the sign. The change is continuous and every intermediate matrix is a basis.
The diagonal matrix now contains ±1 all the way down. If two entries are -1 we can swap those rows and negate one of them, then swap them back and negate the other. Thus pairs of negative entries can be made positive. Finally the last -1, if present, can be moved to the bottom right by another swap swap operation.
The arbitrary basis V has been continuously transformed into the identity matrix, or the identity with -1 in the lower right. The path of intermediate matrices is always a basis. The sign of the determinant never changes, so the identity matrix has -1 in the lower right iff det(V) < 0. Connect W to the same identity matrix, and V and W are connected. Either basis can be morphed into the other.
In the field of complex numbers there are no restrictions on the sign of the determinants. The last -1 on the main diagonal can be morphed into 1 by walking around the unit circle in the complex plane.