It's simple. The passage uses "direct speech", which means it sets out exactly the words that the people used, in the way they used them, so that you know exactly what they said and how they said it. When the people actually spoke, they used the present tense, so it stays in the present tense in "direct speech". English also uses "indirect speech", a.k.a. "reported speech", where instead of copying out the exact words that the people used, you describe or paraphrase what they said. Then, you would probably use the past tense, though at the time of speaking the people would have used the present tense.

Two points: No, the payment didn't happen in the past. They booked the room. They handed the clerk a credit card, and now they owe the money. The payment will actually occur in the future. Also, it doesn't matter that much whether the payment was past or present, or future for that matter. When I say "I am paying...", I do not give you any information about a time frame. For example, I can say "I am singing with the choir next month", even though it is in the future. I have the option of saying "I will be singing...", but it isn't necessary. As another example, I could say "I am studying French" even though I haven't taken a French course in the last few years, so long as I think of myself as someone who studies French. I could use past tense, but it isn't necessary.