Lae'zel is a novice who spent most of her life studying and training, before leaving her creche in order to prove herself. And just like any Githyanki, she is not allowed to enter the Astral Plane until she achieves something of worth outside of it.
There are multiple conversations with Lae'zel in which she talks about her dream of becoming a kith'rak. Which is the highest of honors bestowed upon worthy Githyanki that grants them the silver blades and a dragon to ride, practically becoming Vlaakith's Chosen. It is also the whole reason why in her dreams she sees Queen Vlaakith. It is her greatest desire and ultimate goal in life to become noticed by Queen Vlaakith and be her chosen, as it is her greatest fear to be a failure and a nobody in her eyes. Which is why she becomes extremely pissed off if you act dishonorably towards her.
As for the kith'rak, she is pissed off at him for several reasons.
First is that he broke protocol. An infected Githyanki is supposed to be taken in for purification, but instead he was willing to kill her right then and there.
Second is that he said Queen Vlaakith herself wants the artifact found at any cost, even if it means killing her children. Which to Lae'zel is the ultimate insult, as she doesn't believe her Queen would kill her own children. So she becomes furious and wants to rip him apart for even implying that.
He is not acting according to kith'rak standards nor upholding them, which is why she deems him unworthy of the title and why she promises to rip out his tongue later in the camp. This is all in the game, explained, talked about, in character.
