the red nucleus is present in the midbrain at the level of the superior colliculus.
It is part of a cerebellar connection called the globose-emboliform-rubral outflow. I know you said simplified, so bear with me here. The globose-emboliform nuclei are in the intermediate zone of the cerebellum, which generally deals with control of non postural musculature. So, from the cerebellum to the (contralateral) red nucleus, and then onward as the rubrospinal tract (which crosses), to control non postural musculature - mainly the flexors of the upper limb ipsilaterally.
Here you can tie in the clinical relevance. Lesions above the red nucleus lead to decorticate posturing - upper limbs flexed + lower limbs extended, and lesions below it lead to decerebrate posturing - upper limbs (+ lower limbs) extended.
So basically, the rubrospinal tract is a pathway for the cerebellum to help control the movements of the ipsilateral upper limb (along with the dentato thalamic outflow).