There are several versions of Age of Empires available, so whether or not it is free on PC depends on which version you are referring to.
Age of Empires Online is a free-to-play version of the game that was released in 2011. However, the game was shut down in 2014, so it is no longer playable.
Age of Empires: Definitive Edition is a remastered version of the original Age of Empires game. It was released in 2018 and is available for purchase on the Microsoft Store for $19.99.
Age of Empires II: Definitive Edition is a remastered version of Age of Empires II. It was released in 2019 and is available for purchase on the Microsoft Store and Steam for $19.99.
Age of Empires III: Definitive Edition is a remastered version of Age of Empires III. It was released in 2020 and is available for purchase on the Microsoft Store and Steam for $19.99.
So, while there was a free-to-play version of Age of Empires Online that was available in the past, the current versions of the game that are available for purchase are not free.