/u/majoranaspinor has correctly pointed out that this is due to the GIM mechanism.

Really that comes down to the symmetry breaking in the electro-weak sector.

I don't know how understandable the rest of this will be to you, because I can't think of a nice way of explaining it other than just going through the math. You're best off finding a good reference on the higgs mechanism in the standard model and following that through, but for the sake of try to be expository:

The only reason we have flavour changing currents is because you can diagonalize the up-type and down-type Yukawa interactions (i.e. higgs couplings) separately owing to the fact that the right handed fields aren't in an SU(2) doublet, they are just in singlets.

Neutral current interaction cannot mix up and down types. Doing so would mean they're no longer invariant under the unbroken symmetry, which is electromagnetism, and thus they'd be charged, not neutral. Stated in another way, charge conservation forbids it. So there you know you won't have Flavour Chaning neutral currents that switch up and down types.

Furthermore because Neutral currents don't mix up and down types the diagonalization of the up type yukawa coupling s will diagonalize the up type neutral current couplings and the same goes for the down type diagonalization.

However, charged currents mix up and down types. In diagonalizing the higgs terms doesn't need to diagonalize the charged current terms because the higgs up and down terms are diagonalize sperately. In the standard model the charged current interactions are almost diagonalized when the higgs terms are, but not quite.