Swing speed is about swinging with the whole body and not just the arms. Just like arm speed comes from the lower half and up through the trunk, so does bat speed.
Proper hand position on the bat that allows the hands to take the proper path at the proper angle.
A stable athletic and balanced stance that allows for movement in any direction.
Being in motion vs being static allows the hitter to initiate his weight shift to the back side more quickly while also establishing momentum and loading the swing with potential energy.
A compact front foot stride that keeps most of the weight and energy to the back and optimizes balance provides a stable platform for generating batspeed and power while giving increased separation between the front foot and hands.
Beginning the swing with back hip rotation and building up as much torque as possible between the hips and hands is where increases in batspeed generally can be developed. This separation and torque really set the hitter up for max batspeed and power.
Also be sure to consider the barrel of the bat as the end of the lever and not the hands. The goal is to get the barrel of the bat to the hitting zone as quickly as possible while also keeping the swing on-plane for as long as possible.
Objects move faster if they are furthest from the axis of rotation and the faster axis creates the fastest speeds. Balanced yet aggressive and forceful hip turn are key, but everything else I mention will allow that to take place.