“What a gentleman! Thank you very much.”

You're a woman. You're walking into a building and a man who you don't know opens the door for you. You weren't expecting it, and you appreciate it. You smile and say:

What a gentleman! Thank you very much.

what a (something)

When you say "What a gentleman!" it means "You are really a gentleman." Similarly, if someone does something really nice, you say:

What a nice guy!

This means "He's a really nice guy."

But you can also use "What a ___" for negative things also:

What a shitty day!

This is what you say to a close friend or family member when lots of bad things happen to you on the same day.

a gentleman

A "gentleman" was historically an upper-class, wealthy man in European society. Today, when you call someone "a gentleman", it means that you think he is polite. Because the word comes from old historical ideas, it has the feeling of being old-fashioned. So when people use it, there's usually a small sense that it's a joke.

Unfortunately, there's no word for a really polite woman, although if you tell a woman who holds a door open for you "You're such a gentlewoman", she'll probably think it's funny.

Thank you very much.

"Thank you very much" is a quite formal way to say "thanks" to someone. Because it's so formal, people sometimes use it as a joke in situations that don't really need to be so formal.

In the example at top, the speaker doesn't need to be super-formal to a stranger who opened the door. But since he did something very polite by opening the door for her, she responds with the formal and polite phrase to thank him.

A more common response if someone opens a door for you is just:

Thanks.