On this page of the guide to Crash Bandicoot 4 you will find information about how to unlock the N.Verted levels.
Defeating the boss
In order to unlock access to N.Verted levels, you must first defeat N. Brio. He is the second boss you encounter during the game. He will be waiting for you in the Trouble Brewing level.
Once you defeat him, you will be able to unlock N.Verted levels.
In your guide you will find a walkthrough of the battle with N. Brio. Simply check the page N. Brio - boss fight.
Unlocking individual levels
All levels available before encountering N.Brio will automatically unlock their N.Verted versions once the boss is defeated. You will be able to play them immediately.
As for further levels, in order to unlock the N.Verted versions you must first beat the regular versions of the levels. If you don't complete the regular version of a level, you won't be able to access the special version.