I guess one preliminary thing that I wanted to address is the common use of the well-known acronyms (e.g., common in this thread, LGBTIQA+). Generally, these have to do with diversity in sex, gender, and sexual orientation. While doubtlessly same-sex marriage shows that we have a far greater understanding of those issues, many people within those groups -- particularly outside of the first three letters -- have perhaps far greater issues facing them than marriage equality. Just food for thought.
Almost all law, by the most elementary definition, discriminates. It imposes rights (or obligations) on some, but not on others. For a marriage related example: Only certain, accredited celebrants can marry two people. Others don't have the right (and the obligations that attach to it) to do so. However, it is clear that there is a very good reason for having a register of civil celebrants, and not have just anyone perform marriages. It's a position of civic responsibility.
Of course, discrimination is far worse when you are treating someone differently on the basis of an intrinsic characteristic. It's why we recognise that generally, any discrimination based on them is wrong.
Just looking at discrimination on the right to marry, on that basis of sexual orientation, there simply isn't a logical reason to do so. While you may personally find homosexuality wrong, our society doesn't lend those judgments any credence unless direct harm to another person can be proven. Otherwise, we'll feel merited in removing people's rights based on immutable characteristics, such as their sexual orientation (but also race, religion etc) because we find it morally wrong. People used to find it morally wrong that women vote, for example. And people found it wrong that Jews can own property.
The tl;dr, I suppose, is that you have to come up with an argument better than "ewwww!" or "tradition and history!" or "my religion!" for opposing same-sex marriage. Opponents have been unable to do so, which is why the US Supreme Court struck down laws banning same-sex marriages as unconstitutional.