Had better is a construction called "modal idiom" by Quirk et al. It works like a modal verb, but not exactly.
It works like a modal verb in that it is followed by an infinitive without the particle to:
Travellers had better get their reservation well in advance if they want to fly during winter vacation.
You can substitute it with some other modal verb, it will also not require the use of to
Travellers will get their reservation... (modal verb will: no need to use the particle to)
Travellers must get their reservation... (modal verb must: no need to use the particle to)
There is another modal idiom that works like "had better": "would rather" -
Travellers would rather get their reservation well in advance than wait until the last moment. (see, it uses no to too)
There are two modal idioms that use to: "have got to" and "be to" -
Travellers have got to get their reservation well in advance, or they might loose their chance to fly.
Travellers are to get their reservation well in advance under a new law.
The meanings of these idioms are slightly different but what's important for your question is that only two of them use the particle to.
Reference: Quirk et al., A Comprehensive Grammar of the English Language, Topic 3.45, "Modal Idioms", page 141