I always found that phrase confusing.
What does it mean?
When I have heard that expression used in conversation, it's usually a request for more information or context--a request for clarification. For example, a speaker makes a comment but the comment isn't clear or it doesn't seem connected to what went before or the intent behind the comment isn't clear. So a listener might ask the speaker to "finish his thought"; it's a polite way of saying, "Give me more context on that," or "What's your point?" or "How does that relate to what I said?" or "Where are you going with that?"
Alternatively I might say something to someone, and the listener appears to have misunderstood my point, and so I might interject, "Wait, let me finish my thought!" And then I clarify my previous point by explaining my thinking behind it in more detail.
In this context, "to finish a thought" really means "to finish
expressing a thought" or "provide clarification on a previous (expressed) thought by providing additional context."
Introverts will likely hear requests for clarification more often than extraverts. Introverts may pursue an entire train of thought internally but only vocalize a fragment of it, thinking the rest is self-explanatory or simply forgetting that their listener isn't in their head with them and isn't privy to their thought process. So the person listening to them may ask them to "finish their thought" and provide more of the train of thought that led up to the comment or explain the intent behind the comment.
[Edit:]
A related expression is "hold that thought," as in "Could you hold that thought? I have a call coming in." (I've also heard people say, "Could you finish that thought later? I have a call coming in.")
In each instance, "thought" is really just a condensed or oblique way of saying "the process of
expressing a thought," i.e., talking or expressing an idea aloud.