Developer wants to switch Scrum Team - how to handle this as SM

Why not encourage the Developers to self-organize in such a way that each team has no more than about 10 people or so, and all of the skills needed to build a Done increment of work?

Try arranging a bounded environment, such as a timeboxed workshop, with these rules in place, and facilitate it with this objective in mind.

Have a one-on-one conversation with the developer to understand why they want to switch teams. Is it due to personal growth, conflicts, workload imbalance, or lack of engagement with the product?

Discuss with the Product Owner and other relevant managers to evaluate whether a switch is feasible and beneficial for both teams and the organization.

@Ian Michell: good idea, I will look into that. 
@Dhiraj Kumar: which Product Owner do you mean? The PO of the potentially receiving team or the team they want to leave?

The scrum master is the one who is responsible to decide whether to move the resource out or in. The scrum master first evaluate the team member's skillset, expertise, contribution to the project etc and make sure the project can be move smoothly without the resource. Need to have a team meeting and discuss about the same in the meeting and get the consent from the team members. Then update the stakeholders including the product owner and initiate a new resource request.

This blog post written by the co-creator of Scrum is as relevant today as it was back in 2012: https://kenschwaber.wordpress.com/2012/07/25/self-organization-and-our-belief-that-we-are-in-charge/

From the blog:

Imagine the arrogance of a Scrum Master who believes that he/she knows how developers should organize better than the people themselves. This Scrum Master often has the thought in his/her mind that to be sure, he/she must control the situation; after all, they think that they are in charge (they think).

Depends on whether you are a manager or not. 

If yes then follow a standard process created by HR for internal job/role switching.

If not then well, depends on your context, your speheres of influence and power. 

Don't play manager unless you are a manager, otherwise it's a short way to conflict yourself with other people.

@Chris - I do get that Ken presents the philosophy behind the IDEAL world of Scrum, where reality says otherwise. It's not good nor bad per se, it's just is.

"The Scrum Master has not authority to tell the development team or the product owner what to do their job. The Scrum Master can coach, teach, establish learning situations, enter Socratic dialogue, and parent. However, they weren’t given the authority to manage the other members of the Scrum Team."

I'd say this is dangerous if not entirely unrealistic - budgets are not made out of rubber that can be expanded till it snaps, stakeholders require results, managers have their goals, so on.

What Ken imo describes is some sort of a good fairy that just gravitates around real world people and sprinkle advice-dust, not takin any responsibility/accountability whatsoever.

I mean, um... who actually pays for that?