“Why do local anesthetics cause vasodilation?”

Interfere with sodium and calcium in cell membrane

They block the sympathetic nervous system which maintains vascular tone

Local anesthetics typically block the nerve input to blood vessels in the local area which are responsible for providing the message which cause constriction of blood vessels. If an anesthetic agent is administered which interferes with the messages from nerves in the area to cause blood vessels to constrict, the result is that they tend to dilate. It's not really that the anesthetic causes vessel dilation itself, but rather by blocking the nerve input to constrict, vessel diction naturally occurs.

Vasodilation is caused by sympathetic systemic block (i.e., spinal block).