When English speakers talk about a "gut feeling" or "gut instinct", they mean a feeling about something that can't be logically explained.

The "gut" actually means your stomach; English speakers sometimes think of the stomach as a place where feelings come from. When you worry about something, for example, you may get a nervous feeling in your stomach. The "gut" is also considered to be a place of wisdom. People often tell each other to "follow your gut".

Just follow your gut. I'm sure you'll make the right decision.

You "have" a gut feeling:

I can't explain why I did it; I just had a gut feeling about it.

You can also describe exactly what you're feeling with a clause like this:

I don't know about this deal; I have a gut feeling that I'm going to regret it.

The phrase "gut instinct" is similar to "gut feeling", but it's usually used in a slightly different form:

My gut instinct tells me that it's going to be a boy.

My gut instinct is that it's a boy.