Quick answer: a cheer bow is a large, team-coloured hair accessory worn during practice and competition - partly practical, for keeping hair secure and out of the face during stunts and tumbling, and partly one of the sport's most recognisable visual traditions.
The Practical Reason
Cheer involves fast, physical movement in every direction - jumps, tumbling, being tossed and caught in stunts. A securely fastened bow keeps hair controlled and out of an athlete's eyes and face, which matters both for performance and for safety during stunts.
The Team Identity Reason
Beyond the practical purpose, bows are almost always made in a team's specific colours, and often become closely associated with a gym or program's identity - much like a uniform pattern. Athletes, and often their families, can build up small collections of bows over a competitive career.
How the Tradition Developed
Large decorative hair bows became closely associated with cheerleading over the second half of the 20th century, growing alongside the sport's increasing athleticism and visual presentation. See History of Cheerleading Uniforms for more on how the sport's overall look evolved.
When Bows Are Worn
Bows are typically worn for competitions and performances, and often for practice too, depending on team rules - some gyms require a bow at every practice to build the habit of wearing it correctly and securely.
Related Equipment
See Cheer Shoes for the other unmistakable piece of cheer kit, and A Parent's Guide to Cheerleading for a wider overview of what's typically needed.