What does correcting for LSR mean in galactic velocities?
Proper motions are not velocities. They are angle traversed per unit time. They can be converted to tangential velocities (i.e. tangential to a line joining the star and the Sun) if the distance to the star is known, but you need additional information to get the line of sight (radial) velocity and hence a 3D velocity.

Such velocities are with respect to the Sun (or possibly the solar system barycentre - a tiny difference).

The local standard of rest (LSR) is an estimate of the average motions of stars in the solar neighborhood. The Sun has a peculiar velocity with respect to the LSR. To get a velocity with respect to the LSR, you need to subtract the solar motion with respect to the LSR from your heliocentric velocities.

There are disagreements in the literature (at the level of 1 km/s) as to what the solar motion with respect to the LSR is.