As part of the official team uniform, British players were, and I believe still are given a physical cap to wear, usually embroidered with the match, date etc., not just in football but in wider team sports. The caps are largely symbolic now, and I don't know if they're given out with every match, but certainly they're given one for their first match, at least for a national team. The term's since entered wider sporting parlance.

You can see an example of an England rugby cap still being awarded here.

The term came about in the 19th Century before sports were widely professionalised, and team kits were far from uniform. To tell which player was on which team at a glance, players often wore distinctive headgear, as this could be arranged fairly readily. Up to the mid-late 20th century, most English school uniforms included a cap, widely recognisably in popular culture like the popular Just William books. As these were usually in school colours, they provided a readily available and visually distinctive garment that became the inspiration for sports teams.