Leveling improvements show that these ability improvements are strictly because of training and experience.
With regard to the pre-existing abilities, before level 1, the same can also be true.
In my setting, a level 1 character is an adult, roughly 20 years old, roughly equivalent to a college level student, during the "apprentice tier", levels 1 to 4.
As a teenager, this character has already been training before reaching level 1. There is a kind of "level zero" sotospeak, when the proficiency bonus was presumably +1, not +2 at level 1. Likewise, a time when the Wizard in training could only cast one slot-1 spell per long rest, rather than two at level 1.
The point is, an unusually high ability score can have resulted during this "level zero" training so to speak, while a teen.
Similarly a human spends their feat at level 1, to gain a +1 improvement to an ability score. Depending on character concept, this ability improvement might result from previous training.
Similarly again, player characters can use an array with high-ish scores in the Players Handbook for their characters. Depending on which settings, typical members of the same race might use an array whose scores are much lower. If the player characters are significantly higher than is typical, then the improvements of the player characters might be the result of training.