It indicates to the fans, casuals & hardcores, what you'll find discussed in detail and illustrated through various drills in roughly every modern striking-based combat sports instructional book & video published to date (Boxing, MMA, Kickboxing, Muay Thai, et al). Yes, even Boxercise footwork drill instructionals like Andy Wake's. By controlling the center of the ring/cage you're in an optimal position to control the distance/space and, as a result, often the pace of the fight. It's half of the formula for ringcraft and conventionally the initial precursor for ring generalship (a scoring criterion).
As Kellerman astutely pointed out last night, you can also work farther out around the center's perimeter safely. Controlling the center helps a fighter avoid getting trapped against the ropes or stuck in the corner and looks better to the judges since you're not giving up so much ground that it looks bad. Most fighters with good ringcraft immediately start by taking the center and then work to get back there quickly once they find themselves too far outside or in bad spots. In other words, their spatial awareness kicks in based off of their relative positioning and they'll gravitate back towards the center particularly if they're feeling uncomfortable where they were at. You're taught to do this at the gym in sparring provided that your coach or trainer isn't some hack.
For an out-fighter/boxer/counter-puncher type it's intelligent to work from the perimeter using in-and-out movement, lateral movement to circle tightly and pivoting to stay centered. On the flip side, some "slick" boxers are so comfortable wherever the fight goes that they can fight effectively from anywhere in the ring, even in positions conventionally viewed as disadvantageous, and they'll use that to their favor.
That other HBO stat relating to distance traveled cumulatively over the course of a fight is simply a fighter's foot movement (in steps) tracked to estimate their economy of motion/movement. It can radically differ from style to style, ie., a boxer "mover" vs an efficient minimalist requiring little foot movement, and it decreases substantially as you move up through the weight classes. This stat is solely for scientific boxing purists and its core principle is rooted in physical sports & exercise. Less movement means less waste and that's more economical. Neither stat really belongs, particularly the latter, and "centering" (holding the center) as its own entity isn't significant enough to stand on its own merit as a stat.