In that sentence ‘should’ve made it’ means you started to go home but didn’t get there. It’s a bit odd though. It would be more usual to say I didn’t make it home. I was trying to go home but for some reason I didn’t arrive. I should’ve gone home suggests you didn’t even try to go home.

"To make" is surely one of the most important verbs in English. It has many meanings. One of its meanings is "to arrive somewhere within a specified time". The sentence stucture is: I (subject) should have made (verb, present perfect with modal "should") it (subject) home (adverb modifying verb). Notice that "home", normally a noun, is used here as an adverb. Actually, "home" can behave as any part of speech. It can also be an adjective or a verb.

I should have made it home. Is correct in English,a bit coloquial,it has the same meaning as I should’ve gond home.