In the following sentences, what's the difference between "would become" and "became"?
I'm not sure why this sentences use "would".

This continued after high school, and at a certain point, I began to notice this pattern in myself where I would become interested in an area.

See a translation

"Became" means it happened only once.
"Would become" means it happened many different times.

"I would (verb)" is the past tense, and it means we did something many times, over a long period of time.

"Every day after school, I would pass by the cafe and order a cup of coffee."
"When I was young, I would always run around with my toy bear in my backpack."

Sometimes "I did something" and "I would do something" can both have similar meanings.

"Every day, I passed by the cafe and ordered a cup of coffee."
"When I was young, I always ran around with my toy bear."
These sentences also work.

"I would do it" is usually considered to be less specific about the time and frequency. You're saying that you did something frequently, but it's not very important what the specific time was, or how many times you did it.

"I did it" is usually considered more specific. There was an exact time when you did something, and you know exactly how many times you did it.