How do you explain large amounts of something like たくさん?
I saw thousands of, hundreds of and twenty times...
I'm really curious about how they choice that kinds of words to explain quantities.

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How do you EXPRESS large amounts of something like たくさん?

You could say:

"tons of"
"There were tons of people in the city"

"a lot of"
"There were a lot of pigeons in the sky"

"loads of"
"Loads of stuff are on your way. Look out for it."

We don't often express large quantities of something as "millions". Instead we use the terms above to indicate that there is too much to describe in numbers.
We also estimate quite a lot thus the terms "a lot of" and "tons of" are used quite often. You can still use "millions" but it doesn't sound as natural as using terms like the ones mentioned above.

"There were millions of spiders in my house".

Also, たくさん is often translated as "a lot of" or "quite a few"
このクラスの中にテストをしっぱいした人がたくさんいた。
"There were quite a few of you who failed the test"