Most people are desperate to be listened to and understood, and when they feel like they're not being understood, they talk louder. Break out of the doom loop by developing active listening skills, and learning how to validate other people's opinions.
Often times once a person feels listened to, they stop worrying about "being right" and are willing to make compromises, or even drop their demands entirely.
A great way to help someone see why you think their solution is wrong is to ask questions. "What happens if the user clicks X?" "How will the database handle the increased scale?" "What will be the costs of implementing that?" etc. etc. Just pure, calm curiosity.
But yes, even if you're in a debate where the sole purpose is to win, listening is important. Look at politicians during political debates -- they listen like hawks when it's not their turn. The reason is simple: you can't effectively negate an argument you haven't even heard. We're all smart people and capable of coming up with a million counterarguments to everything, but hearing EVERYTHING someone says is the really hard part, and is worth practicing.