The consequence of divine simplicity is that God is being subsisting and can only be immutable.
If so how does God do anything? Or how did he create?
Per St. Thomas Aquinas (and Aristotle), things have actual states and potential states. God, per Aquinas, has actual state but no potential (https://www.scotthambrick.com/article/1081/). This makes Him immutable. This is part of divine simplicity (https://pintswithaquinas.com/what-is-divine-simplicity/).
Other, contingent beings have both actual and potential traits. In order to create change in a contingent being, something has to activate that potential. Ultimately every potential that's activated must result from some cause that is not itself caused by something else -- like if you're pushing something with a stick, the motion of the stick is caused by the motion of your hand is caused by the contraction of your muscles, etc. If there's no prime cause, there's no motion. So there has to be an ultimate cause that is itself not contingent, that is not having its potential actualized by something. "This," Aquinas says, "everyone understands to be God" (https://home.csulb.edu/~cwallis/100/st2.html).
I wouldn't know how to address a perceived contradiction between being immutable Oneself and causing change in other things, because like other commenters and AFAIK Aquinas I don't see the contradiction, but it's clear Aquinas thinks that every change in our world requires a simple and immutable Cause.