Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation}
It is a well-known fact that $W(T)$ is a convex subset of the complex plane. However, every proof I know is by brute force computation. First for $2\times 2$ matrices, then the general case.
Even though the computation can be carried out in clever ways, it still fails to provide some explanation why this is true. What is the link between this result and other concepts of the theory?
I wonder whether there is any conceptual explanation for this result. I do not ask the explanation to be rigorous, just some ideas.