Without "having to", the sentence would be:
"Love means never saying you're sorry."
"Love means you never have to say you're sorry."
Both of which are untrue in the specific sense. People in love say (and need to say) "I'm sorry" all the time. The true meaning of the quote is, on a general level, that even if you do say you're sorry it doesn't absolutely need to be said, as people in true love will always forgive each other.
"Love means never having to say you're sorry."
Using the generic "having to" rather than the potentially specific "you have to" helps elevate the statement from specific-to-this-apology advice to a general proverb about relationships.
Proverb: A short pithy saying in general use, stating a general truth or piece of advice.
It's true that the pronoun in "you have to" can be taken as a generic you, which gives it mostly the same meaning as "having to".
But the (implied) subject in "having to" is an even stronger indefinite pronoun than the "you" in "you have to", which could easily be taken as a specific singular "you" subject.
Using "having to" definitively indicates that the phrase is a general statement that applies to many people, not just those at hand.