こんにちは (Konnichiwa) Michi Tsubasa,
Thanks for your question!
This way of using の (no) is a little different to what you might be used to.
When の (no) comes before です (desu) like it does here, it is used to express a reason or provide/ask for more information. You can read more about this useage of の (no) in this lesson:
The の (no) here implies that the speaker (Kenny) wants to find out more information about booking a room. の (No) doesn't really have a translation in English, but it hints at wanting to know more. So, you might translate this sentence as something closer to:
“(I) would like to book a room (so please tell me if a room is available).”
Your understanding of へや を よやく したい (Heya o yoyaku shitai) ”(I) want to book a room” is pefect! However, です が (desu ga) does more than just shape the sentence. The が (ga) here is used as a polite sentence ender to soften what Kenny is saying. “(I) want to book a room” sounds a little too direct in Japanese, so we often soften this by adding ですが (desu ga) at the end. This changes the meaning to “(I) would like to book a room” - much softer, right?
We're currently working on writing new Interactive Audio lessons for our Japanese course, which should have much clearer explanations on what each part of the the sentence means.
I hope this helps.
Please let me know if you have any other questions.
べんきょう を がんばって ください！ (Bekyō o ganbatte kudasai!)