When I first spent significant time in London, about fifteen years ago, one of the first words that struck me as unusual was book–used as an all-purpose verb to indicate, well, as the Oxford English Dictionary puts it: “To engage for oneself by payment (a seat or place in a travelling conveyance or in a theatre or other place of entertainment).” The OED cites a first use in Disraeli’s 1826 novel Vivian Grey –“I’ll give orders for them to book an inside place for the poodle”–and then an 1887 theatrical advert: “Seats can be booked one month in advance.”
In London a hundred-plus years later, people were always talking about booking theater, excuse me theatre, tickets, hotel rooms, tables at restaurants. (Granted, this came up a lot because I was in a tourism/study abroad situation.)
The usage was not unheard of in the U.S., but (the significantly less strong) reserve was much more common in reference to eating and sleeping. For theatre, I guess our best word has been get, which isn’t very good. Hence it’s not surprising that book has gotten traction here.
This Google Ngram shows the change in use of book a room (blue) and reserve a room (red) in American English, 1970-2008. Book surges ahead in 1993, which puts it in a sweet spot for NOOBs.