The reasons for applying black paint, "eye black", or (these days) glare strips to an athlete's face have evolved over the years.
The original inspiration, and still a valid reason today, for the application of dark substances under each eye is to reduce glare reflecting from the Athlete's skin into their eyes. However, such application is only necessary for athletes playing particular positions and in particularly harsh (i.e. bright) conditions. Some studies (as well as, at least in part, the continuing popularity of the products) suggest that blacking out under the eyes can provide some relief. In today's world of competitive sports, athlete's will go to great lengths for the smallest advantage, so it is no surprise to see the no-glare practice perpetuated.
However, glare is not the only reason athlete's apply eye black these days. There are two additional reasons-one inspirational and one practical-that more and more athletes have adopted the practice, whether they need it or not.
Having played major college football in the Big Ten Conference in the 90s, I fondly recall the pregame (and even pre-practice) rituals we all went through to physically and mentally prepare for the beatings we were about to give and receive. Each of us had our routine and superstitions. Some of us liked to liken ourselves modern-day gladiators preparing for battle. As part of that, for some I suspect, eye black took on the mystical properties of war paint. We were marking ourselves for battle, and it was really the only acceptable marking that we would be permitted to do.
Other writers have opined that athletes where eye black to intimidate each other. I think that might have been the case a few decades ago, but with the prevalence of the practice these days, there isn't a whole lot of intimidation going on. Anything more than a simple strip under the eye is just viewed as plain silly. Today, it seems, the mohawk has been adopted as the intimidator de jour. Tomorrow, it will be something else.
More on the practical side is the use of eye black to distract opposing players from looking at your eyes. Any American football defender will tell you that one of the tips they are taught is to follow the quarterback's eyes, as the eyes will always tell you where the play is forming. Of course, the same coaches who were teaching that to their defenders realized that opposing defenders would be looking at their own quarterback's eyes as well. Several techniques were adopted to combat this practice. One of them is the faceguard shield. The other is the imprinted glare strip.
A glare strip is just a more modern version of eye black. It's a black adhesive strip that sticks to the skin under the eye. One benefit is, unlike other materials used to black the eye, the strips won't smudge or come off when you towel off sweat. Another advantage is that they can be imprinted with something. Some companies have taken to imprinting a light object in the center of the black strip, giving the appearance of a shadowed eye. The idea is that in the heat of battle and with only an instant for a defender to gain any information, the fake eyes will confuse and/or send the wrong information.
The practice of using eye black or glare strips is still prevalent in many sports, particularly American football. The original purpose of reducing glare has, however, certainly taken a back seat to more trendy (pregame preparation and expression) and deceptive (those second eyes) purposes.