Hello! Yes, it is true! Basically, parts of the motor cortex on the left hand side of the brain control the movements of body parts on the right hand side of the body. This is also true of vision: the left hand side of the visual cortex activates in response to information presented to the right eye.
The way this works is that the neurons going from motor cortex to the body cross over on the way down (it’s called “decussation”) so that they connect (eventually) to muscles on the opposite side.