If leftist: I believe that oppressing others for my own personal gain is wrong, there is no consistent framework which excludes animals from moral consideration that does not also exclude some humans. Using intelligence to rank who you're allowed to hurt is Nazi shit.

If not leftist: you know how watching someone hurt an animal makes you sad? Killing animals to eat hurts them.

Anyone who is trying to argue with you about your own personal reasons for being vegan is just not really worth speaking to lol.

I mean, usually these questions aren't posed, because they're genuinely interested, but rather because they feel attacked.
And then, yeah, there's no point in delivering a logically flawless argument, they won't actually listen anyways. The only thing they'll listen for, is whether you're attacking them or not.
Formulating I-messages signals that you're not attacking them, which is probably better for resolving that situation.

Having said that, I often don't heed that advice either. I usually don't want to talk about it. If they bring up the topic and want to feel offended, I have no problem making them feel offended (by stating harmless facts), so they leave me alone next time.

More often than not, questions comes from people trying to pick a fight in my experience. But since I still want to, at the very least, make them consider reducing their meat consumption, I talk about money: how going vegan made me save a third of my previous grocery budget, taking into account all the fancy spices I've started buying since then.
I've had some success with this.

Liberty for all means all