It is "would rather". We use it to express our preference. "I'd (would) rather stay in tonight." Just as a side discussion: If the writer wrote: "I'd better stay in tonight.", then I'd is a contraction for "I had". "had better" is used to provide a strong advice or warning. In this case, the writer was reminding himself/herself not to go out.

I would rather stay in. :)

It's "would" because "stay" is in the present tense. "Had" would have been "stayed". I don't know if that makes sense :-)

i Would like to stay in tonight, because the weather is bad. or I don't wish to go out tonight, as the weather has become worse ...