Also too: in the cultural context of Westeros, “breaking lances” can be shorthand for “is unexpectedly badass.” Like, take Ser Arlan in The Hedge Knight:
Well, a tourney lance is made to shatter rather than impale your opponent, because tourneys are supposed to be nonlethal. So breaking a lance meant that you scored a good direct hit against your opponent’s shield or armor.
See, the thing is that while knocking the other guy off your horse was a clear victory, jousting didn’t always require it. There were quite a few victory conditions, depending on the style of joust – knocking off the crest of your opponent, causing a spring-loaded shield to detach, or simply winning on points.
To use a modern analogy, winning a boxing match by K.O is very straightforward, but a lot of matches are decided on points. So breaking a lance would be like landiing a hard and clean shot which will contribute to the judges scoring a round in your favor.
Ser Arlan had not ridden a tilt since the day he had been unhorsed by the Prince of Dragonstone in a tourney at Storm’s End, many years before. “It is not every man who can boast that he broke seven lances against the finest knight in the Seven Kingdoms,” he would say. “I could never hope to do better, so why should I try?”
In this context, and in many other places besides in the books, “broke lances” means “I rode a tilt against a very skilled, dangerous opponent, and broke my lance against him, and was not defeated in that pass. And then I did it six more times!”
So when someone refers to “breaking their lance” in a positive sense, they often mean “I was not defeated and also acquitted myself well.”
Yep, think of it like Rocky saying he “went the distance” against Apollo Creed.
On a somewhat related note, the idiom of breaking one’s lance was used more broadly than merely jousting. For example, it was noted (in a letter to the Pope no less!) that at his wedding, Cesare Borgia “has broken four lances more than he, two before supper and six at night, since it is the custom there to consummate the marriage by day.”