It depends.
In a magical world I expect people to have evolved to have a biology that is capable of interacting to some degree with magic. Most have a limited ability and can never get beyond the basics - thus part of the reason for many of the lowest level magic using NPCs. They literally cannot get further along even if they were to train for decades, adventure, etc.
Some have greater ability. While it might be called a 'spark' it is more accurately referred to as either heritage (1, 5, 10, 50, etc generations back, one of your ancestors was a fey, fiend, celestial, etc) or the blessing of a powerful entity (deity, fey, fiend, etc).
In the former case the caster has an innate ability which might have revealed itself through minor incidences of wild magic, odd luck, etc that some trainer in the arts recognized, leading to the start of training. Or perhaps the nacient mage was studying old tomes and found that some of the spells / etc in them worked for him despite not working for others.
In the latter case, a dream or a series of strange coincidences might have lead to a potential cleric, druid, etc stumbling upon a high master of the arts and thus becoming a student. They may not have been born with the ability (at least not to any great degree), but their minor innate ability was expanded greatly by some entity with an interest either in the individual or what the individual might do for them in the future - if suitably empowered and trained.
