Here's why:
* Subject: A complete sentence needs a subject (who or what is doing the action). While "rain" is a noun, it's not clear who or what is causing the rain to fall.
* Verb: A complete sentence needs a verb (the action being done). "Fell" is a verb, but it's not clear who or what is performing the action.
To make it a complete sentence, you could add a subject:
* The rain fell. (Subject: The rain)
* It rained. (Subject: It, referring to the weather)