Why can we measure current in AC voltages?
What am I missing here?


AC voltages cause no \$\color{red}{\boxed{\text{net}}}\$ movement.

You might have missed the red word above. It means average i.e. there is no long-term average movement but, of course, there is movement of charge for each half cycle of the applied AC waveform; one in one direction and the other in the reverse direction, averaging to zero.

If AC voltages are causing no movement how is it that we can measure
current

AC is causing movement but no \$\color{red}{\boxed{\text{net}}}\$ movement.
Note to Neil: \$\require{cancel} \cancel{cancel}\$ works in answers