Why is Mr Darcy so silent with Elizabeth?

There are several places in the book where this is explained to some extent.

Darcy is reticent with strangers. He does not feel confident to interact with them, especially dancing, but also in conversation. He is quite capable of interacting with his friends, as Col. Fitzwilliam notes.

In addition to this, Darcy is particularly attracted to Elizabeth, but considers it his duty to his family not to get involved with her. He limits his contact for that reason, especially during his stay with Mr Bingley at Netherfield at the beginning of the novel. Once he has overcome the internal conflict, he can talk with her, but he proposes, is rejected, and has to overcome the hurt to some extent before he can deal with her comfortably. This he does, but only after he and Elizabeth meet at Pemberly, which is Darcy's home ground, where he feels most comfortable.