What exactly happens at the end of a season in Diablo 3?

Summing it up, drawing inspiration from this wiki page:

Heroes

When a Season comes to a close, all the progress you’ve made with your Seasonal heroes, including their inventory, shared stash, and Paragon experience, will seamlessly merge into your non-Seasonal profile, whether it’s Normal or Hardcore. All the items that your heroes have equipped or stored in their inventory will gracefully accompany them in this transition.

Farewell to the heroes’ seasonal labels.

Shared Stash

The contents of your Seasonal heroes’ shared stash will be sent to your Normal/Hardcore non-Seasonal profile through mail. To access these items, simply enter a game with any non-Seasonal hero and click on the mail icon located at the bottom left corner of your screen.

The belongings you acquire during the Season will not be directly added to your regular storage. You have the option to create space before claiming them, in case you need to. As per the current system, any items sent via mail from your Seasonal storage will be accessible for a period of 30 days after the Season concludes.

Paragon Experience

Players need not exert any extra effort as Paragon experience earned throughout a Season seamlessly carries over.

Achievements

Unlocking an achievement on a Seasonal hero that remains unattained by a non-Seasonal hero will grant you credit for both triumphs simultaneously.

Gold and blood shards

Your non-Seasonal profile will seamlessly receive the transfer of Gold and Blood Shards from your Seasonal profile.

As you transfer your Blood Shards from the Seasonal profile to the non-Seasonal one, the temporary disregard for the 500 cap comes into play. Consequently, players will find themselves unable to acquire additional Blood Shards until their count plunges below 500.

Artisan levels and learned crafting plans

At the conclusion of the season, the elevated level of your Seasonal Artisan shall seamlessly carry over, along with any exclusive recipes that are absent from your non-Season Artisan.

Legendary gems

In the existing layout, should you happen to acquire a Seasonal legendary gem in Normal mode that you already have, you’ll be fortunate enough to possess a splendid pair of these gems.

Rewards

According to the handy in-game guide, reaching level 70 will grant you access to coveted Transmogrifications. Additionally, there’s a chance to unlock unique Banner styles.