Correct spelling, explanation: twenties is a noun used for specifying an age or an era. It’s made based on the number twenty and a plural suffix s, and in the process, the y is changed to ie. This is because of a principle in English, that when there is a consonant before the y, it has to be changed to ies, and in twenty it’s t, which means that the correct plural is twenties.

Definition of twenties:
noun
1. the spectrum between twenty and thirty usually used to describe age or temperature
My mom was in her late twenties when she had her first child.
In Centigrade, the twenties are the room temperature, but in Fahrenheit twenties are freezing cold.

Incorrect spelling, explanation: twentys may seem like a logically constructed word, but it’s incorrect because of a small principle when making plural nouns. Namely, if the term ends in y, and the letter before is a consonant, the y has to change to ie. This is why in the case of the word twenty, the correct plural form is twenties.