The person has responded honestly, from their "gut feeling", meaning by their intuition and from the facts they have, even if the answer is not absolutely true. Their answer is their opinion, and they are probably very convinced and emotional about their response. If they have more information, they could change their mind, but as it is now, that's the way they feel about the matter.

"that's my gut feeling " = my intuition tells me that.........

You have an internal notion that what you feel to be true is. It does not need to be based on fact, it is a feeling of truth. In my culture, it is too often seen as nobel to make decisions our 'gut' rather than our mind.