I'm not entirely sure how it actually works in terms of code, but what I understand is this: when it says break, it means damaged. Your weapon has a chance of being damage when in use on the battlefield (I noticed my weapons only get damaged when I strike a tree or a unit that is blocking). When your weapon gets damaged, it gets taken down a modifier. So if I had a Balanced Sword, and it got damaged, it would become a Heavy Sword. If it got damaged again, it would become just a Sword (but blacksmiths refer to it as Plain). If it is damaged yet again, it becomes Chipped Sword. Then Rusty Sword. If you reach the final negative modifier for a weapon and it gets damaged again, the weapon is broken permanently - as in removed from your inventory.
To repair items, you only have to go to any town's weaponsmith and ask them to refine your weapon. Refining weapons costs peningas but every time you refine something, it goes up the positive modifiers. So refining a Rusty Sword once gives you a Chipped Sword. Refining it again gives you a Sword. Then Heavy Sword. Then Balanced Sword. Then Tempered Sword.
And like I said I don't know how the Weapon Maintenance skill actually works, but as every level makes your weapon 7% less likely to break, one can assume that at level 10, your weapon is 70% less like likely to break than if you hadn't had any points in the skill. I don't know what the base chance to break is, nor if the nature of the strike affects it (like a powerful high speed bonus strike from horseback getting blocked), nor if the chance is specific to the item you are using. Since Weapon Maintenance is also a party skill so I don't know if it also affects the breaking chance for all the troops in your army. It should, though.
Very commonly when you fight a load of bandits, you'll see yellow sparks fly out of the fray occasionally - this is a weapon breaking in the hands of a troop, leaving him defenseless. I think bandits commonly carry poor quality equipment, as that is what you tend to loot, so that's probably why it happens so often with them.
Hope that answers your question.
Armour can also get damaged, but it's incredibly uncommon, at least in my experience.