If you receive an offer from a different team, how do you communicate to your manager that you plan on moving to a new team / opportunity at Microsoft. What if you don't have the best relationship with them?
At Microsoft, the current manager does not need to approve team switch, regardless of whether it's a software role or not.
Frame the new team as a compelling opportunity, allowing you to work on technologies, projects, or people that would be difficult otherwise. It's better to talk about the attraction of what you're going to, instead of the negatives you're leaving behind.
Oh wow, I didn't know that your current EM doesn't need to approve of a team switch. That's pretty good as it allows engineers to escape toxic managers more easily.
When you're leaving, your goal is simply to not burn bridges. You are heading to a (hopefully) better place, so there's no need to be mean and blow things up on your way out (i.e. just take the win and go). Communicate kindly, and if you don't like your manager, you may need to stretch the truth a little bit haha.
Check out this related thread: "How can I gracefully leave my job?"