On its own, "Lui aggiunge" is not much of a sentence.
Reverso is a helpful website for verbs in context. An example given:
"Lei aggiunge tensione ed emozione ad una situazione già difficile."
She adds tension and emotion to an already difficult situation.
"Lui aggiunge qualcosa al loro calendario."
He adds something to their calendar.
I'm trying! I really can't make up a situation where this works as a sentence on it's own.
there are people discussing the pros and cons of an upcoming move. one person, let's call him larry, has been very quiet the entire discussion but has made some valid points. someone calls him out saying that he isn't adding anything to the conversation, when someone stands up for him-- "he adds!"
There is a list of numbers on the board, and the guys stands up and puts the sum at the bottom. "He adds!"
Mathematical addition is 'addizionare' . Aggiungere is more like mixing something in or merging.
Also it's just showing how to conjugate the verb, it doesn't need to appear by itself in everyday conversation