How does scaling affect Logistic Regression?
It affects anything optimized by a form of gradient descent, because it affects the relative scale of the dimensions of the input. If A is generally 1000x larger than B, then changing B's coefficient by some amount is in a sense a 1000x bigger move. In theory this won't matter but in practice it can cause the gradient descent to have trouble landing in the right place in B's dimension.

I think the more significant effect may be regularization. All terms are penalized equally in a simple formulation of regularization. Again if A is generally 1000x larger than B, its coefficient will tend be 1000x smaller (all else equal) than B's, so will be far less penalized, when there's no particular reason to treat them differently.

Finally it affects interpretation, if you want to view the coefficients' magnitudes as correlating with importance, for the same reason.