A phrasal verb is just a verb plus a preposition or adverb which creates a meaning different from the original verb.
Phrasal verb’s origin is probably as far back as the language is concerned. Any verb would require either an adverb or a preposition to modify its meaning.
For example, when you use the verb “look”, you will sooner or later need to use it with an adverb to modify its meaning.
Through frequent usages, the combination of the original verb with either an adverb or a preposition began to take on additional meanings.
For example, “look up” is “direct your gaze upward”. Then somebody went to look for a word in a reference dictionary, and that very thick and heavy dictionary was placed on a pedestal, and the person literally had to look “up” at the dictionary. Soon look up became associated with finding a word in a reference dictionary or just a plain dictionary.
Or when that person went to see or visit his friend. He didn’t know the address, and he had to look up at the numbers posted near the top of the door. Soon, look up became synonymous with “visit” => I always look up an old friend whenever I stop in Paris.
Or someone would equate the action of looking up as a positive sign. Things are starting to look up => things are starting to improve.
The same story could be told about “sign up”.
When you sign, you just put down your signature. But when you sign up, you don’t just write down your name, you actually volunteer for something that you are really interested in.
Up => with a greater intensity, completely, so as to advance, increase or improve.
Are you up for it? Therefore, you sign up for military service, you sign up for a dancing class etc.
There are equivalent verbs to sign up such as enlist for military service, and enroll in a dancing class. But the beauty of phrasal verb is its colloquialism and its simplicity. One can just add an adverb or a preposition to a common verb to get completely new meanings, sometimes related to the original verb, sometimes not.