Help understanding the Casmir Element in Humphrey's Lie Algebras

Question

I am currently reading through Humphrey's Lie Algebra book and came across the Casmir element. I am having trouble intuitively understanding it. For example, he states that if we have some irreducible representation $\phi: L \to gl(V)$ then the Casmir element is a scalar equal to $dim(L)/dim(V)$. I understand that it must be a scalar by Schur's lemma, but not why it is specifically equal to $dim(L)/dim(V)$. I also do not quite understand why the Casmir element is basis independent, since we needed to construct a basis to define it. Since I am at a roadblock in terms of proving such simple facts about it, I would also appreciate some motivation for why we care about the Casmir element.