I don't know what "MIT [doesn't let] you change the license" is supposed to mean.

You cannot take code licensed under MIT, delete the MIT license, and supply a different license instead.

This is permitted by the terms of the MIT license.

It is permitted to include MIT licensed code in a project that also contains Apache licensed code, but it is not permitted to change the MIT licensed code to be Apache licensed.

The result is a project where different code has different licensing.