Technically "excite" is a transitive verb and therefore should be followed by an object. However, I have seen other examples following similar structures as such:
The little engine that could.
or
The ties that bind.
Could it be an idiomatic expression that follows a Subject + that + Verb model? And if that's the case, does it apply to every verb?
All of your examples are noun phrases, not sentences. To be a sentence, the noun phrase requires a main verb:
The brakes that bind are the green ones.
This book by Arthur Ransome is the story that excites.
With that proviso...
That in these sentences or your phrases is a relative pronoun, referring to the main noun, story, engine, tires, ties, brakes. Any action which the main noun can do can follow that.
This story excites [people]. It is the story that excites.
Technically, "excite" is a transitive verb
No. Technically, "excite" is an ambitransitive verb, i.e. if a direct object is not stated, it can be implied.
"The stories do excite [the reader] but not in a good way."
The little engine that could.
There is an implication here of "do what was required because of determination." (It is a title and titles should never be taken as perfect examples.)
The ties that bind.
again this is ambitransitive. The generally implied object is "people".
Also "Put some glue on it, and it will bind."
