Hi, may I speak to David?

- Just a moment! I'll get him.
---------------------
In this telephone conversation, what does 'get' mean? Bring, catch, etc?

It's casual speech saying I'll notify him he has a phone call.

In this context, the word "get" means to bring. In the conversation, the person who is answering the call is informing the caller that they will bring or fetch David to the phone. The word "get" is commonly used in this way, especially when someone is being summoned or asked to come to a specific location or situation.