You can explain it in the same way people explain why one person has a higher IQ than another, and why another thinks faster than another. It is related to genetics.
Just like genetics can determine whether a person is color-blind, red-green color-blind, "normal", or can actually see more colors than the average person (like this woman), genetics can determine what powers a person can tap into.
Then consider that we only use a small percentage of our brains. Those with magic would simply be more intelligent and be able to use more of their brain power, potentially unlocking abilities that normal humans can't do.
Then you add training on top of that, and you can make someone truly magical. For example, some monks have so much control of their body, they can control their blood pressure, body temperature and heart rate through thought alone. Take this to the next level, and people can start controlling additional things with their mind, even external things.
Then you can add in the metaphysical. 227
Some say that we shape our world with our words and thoughts. Someone who is powerful with their words and mind can will things to happen, and they happen.
So you could explain that a certain level of intelligence is needed to fully take advantage of the power of the mind, and this power of the mind is often looked upon as magic by the rest of the population. Their mind is so strong that they can literally will into existence what they want, including the ability to fly on a broomstick.
<PERSON> might be an important cut off point, because before puberty, humans are usually dependent and curious. Some studies even suggest that children ages 0 to 7 are actually in a hypnotic/dream state where dreams and reality are mixed, and they are very open to suggestion and training. As they reach puberty, they start becoming more independent, thinking for themselves instead of absorbing everything around them. This transition from a being that focuses on absorbing information & learning to one that thinks for itself and thinks independently could lend itself to puberty being the time of selection and screening. 494