What should I (a professor) do if a student keeps answering other students' questions before I start explaining?

This would, I think, depend a lot on how the "answering" student goes about it. Just interrupting a request to the professor is disruptive, but some people do this sort of thing by reflex. But yes, a quiet word is in order. Ask them to meet with you. But decide first on what you would like them to do to correct the action.

If the answers they give are generally correct then it is a different situation than if they are mostly irrelevant, I think.

One thing that such a student fails to understand, of course, is that often the best answer is a minimal answer. An answer that lets students develop insight, not just get answers. Saying too much in answering a student question may be worse than saying nothing.

One option is just to require them to remain silent, but another is to ask (require) them to raise a hand before saying anything and first getting permission. But yet another, that might be useful, depending on the student, is to require them to write out an answer when they feel the need, rather than to blurt it out verbally.

But the behavior can be a symptom of many things. Some are insecurities on the part of the student; a need for affirmation. Some would just be showing off for the professor. Without knowing what it is, it is difficult to formulate an exact solution.

You have a right, of course, not to have your course disrupted by outbursts, but think a bit about whether the disruptor has needs that also need to be met. So, as part of your quiet word with them, ask first why they think it important to do that. Depending on your view of the student, the question might be direct, or you might want to be a bit more subtle about it.

And sometimes it is actually useful to have students volunteer answers to the questions of others as long as you maintain control over the process. If one student is dominating a conversation, it might be counterproductive for others.

There are two main components to be handled:

1. The question
2. The answer

For (1), you can either choose to say

Please ask your question after I finish this part

or take the question right away.

For (2), I usually prefer to ask the classroom

Is there anyone who wants to answer this question?

If someone volunteers, and in your case someone always does, then I let the one who asked the question listen to the answer. Then, the important part comes:

Did you understand the answer?

or rather, as @Bob Brown pointed out,

Is there anything I should explain in more detail?

If the answer is understood, then everyone is happy. If not, then I proceed to explain the answer in my own way.

At the end of the day, the most important thing is that the students understand as much as they can, regardless of the person explaining. Of course, it is sometimes annoying for a student to interrupt your answer arrogantly. But you can always turn this situation into your advantage. Remember that there will be a question where someone else wants to answer, and then your lecture becomes interactive. Interactive lectures, in my opinion, are the best to follow and the most enjoyable ones.

In many classes, there are a few students who participate way more than others. I would recommend doing two things in this case.

1. Have times when it is appropriate for students to answer each other's questions. It wasn't clear from your question whether your class has that, but if it doesn't, consider adding it. Everyone benefits -- some students get their questions answered, others get to think through the material more deeply, and everyone realizes that they can think and learn on their own, without everything coming from you.

2. Make the student into an ally. Pull them aside after class and say that you appreciate their enthusiasm and engagement. Then, make your request -- in your case, say that you need a chance to answer student questions without them jumping in. Don't sound like you're asking for a favor, just making a polite request, preferably with a reason. "I need you to not jump in when another student asks a question. It's important that I can address some subtle issues that can arise." End on a positive note, maybe appreciation or recognition of their interest.