Upward-facing dog pose is a backbending arm support in which it is important to create evenly distributed extension throughout your whole spine. To accomplish this, the thoracic region needs encouragement to bend backwards, while the more mobile lumbar and cervical regions need to be stabilized. This translates into active shortening (concentric) work in the extensors in your thoracic spine and active lengthening (eccentric) work for the flexors in your cervical and lumbar spine.
As in many poses where your arms are active, the latissimus dorsi are not so helpful, because they can fix the scapulae on the rib cage and inhibit extension in the thoracic spine. They also produce internal rotation of the upper arm (humerus) and downward rotation of the shoulder blades (scapulae), both of which oppose the actions of upward-facing dog.
Throughout your spine, the extensors are active, though mostly in the thoracic region. Gravity creates much of the extension in the lumbar spine all on its own, so your abdominals (and maybe your psoas minor) are actively lengthening (eccentrically) to resist too much arching in the lumbar spine (lordosis). In your cervical spine, gravity, acting on the weight of the head, creates extension, so the anterior neck muscles work eccentrically to keep the action balanced. In some variations of this pose, a chin lock is maintained, which makes the thoracic extension that much more powerful, and asks the back of the neck to lengthen as the chin is tucked down towards the top of the sternum.
As the counterpose to the more “exhaled” nature of downward-facing dog, this pose is clearly related to the expansive action of inhaling. Many who practice Ashtanga-based sequencing mostly hold this pose for only half a breath, as they move through it between stick pose and downward-facing dog. By holding it for several breaths, you can allow the action of the inhale to deepen the opening in the front of the ribcage, whereas the exhaling action can assist you in stabilizing the lumbar and cervical curves.