Gibbs-White was a second-half substitute in England's Nations League win against the Republic of Ireland on Saturday.
He said, "‘It feels much, much better than I expected. The boys have been absolutely class with me, made me feel really welcome.
"I can't find words to describe. Ask me in a week and I will probably be able to tell you but I can't really describe the feeling.
"I'm over the moon, especially as the boys got the three points as well which is what we're striving for."
On caretaker coach Lee Carsley, Gibbs-White added: "He's a very hands on manager on and off the pitch. I believe that shows he believes in his own ideas and us seeing that makes us believe in him. It's obviously a good thing when you're hearing it directly from the manager on and off-the-pitch.
"I think the boys are really buying into him, really enjoying the time now and hopefully it can stay that way."