In the interest of not making this answer excessively long, this doesn't cover every single detail, but it should catch you up to speed on the main events and motivations.
After the events of the base game, you meet a mysterious girl named Ansha and her companion Hoopa, who ask Team MZ to help gather ingredients so she can create a donut. After doing so, she reveals that she can feed it to Hoopa to open up strange portals ("distortions") that lead an otherworldly version of Lumiose City, Hyperspace Lumiose.
6 months ago, Ansha witnessed a Rayquaza in Hyperspace and wishes to find and capture it as a gift for her mother. Meanwhile, the Rust Syndicate has begun leading the investigation of the sudden appearance of Hyperspace distortions (as Quasartico is busy dealing with the fallout from the events of the base game), including a massive one above Prism Tower. Corbeau tasks Team MZ with surveying Hyperspace with the newfound help of Ansha and Hoopa. He also enlists Korrina, who becomes another new member of your team.
Hyperspace contains wild Pokémon above level 100 that don't normally appear in Lumiose, dreamlike apparitions of people, and more Rogue Mega Evolutions. Most of the DLC consists of Team MZ exploring distortions and defeating Rogue Mega Evolutions as Corbeau reports on an increasing risk of Hyperspace Lumiose expanding out of control and taking over the real city. By finding high quality butter in Hyperspace, Ansha is also able to create donuts that imbue stronger powers. We learn that Ansha is the daughter of Champion Diantha, who is out of town for work. Team MZ also suffers from nightmares involving a threatening voice.
Many of the other characters from the base game variously help or hinder your survey efforts, but don't play a major role, and there are some scenes of Team MZ hanging out that explore their feelings as well.
Eventually, it's theorized that Hyperspace Lumiose is somehow created from the dreams and subconscious feelings of the inhabitants of the city, and that a single Pokémon is behind it all. By creating a special donut called the Bad Dreams Cruller, Team MZ is finally able to enter the huge distortion above the tower and arrive in a forested location that looks like Newmoon Island. Here, an apparition of AZ appears briefly. You then confront Darkrai, the source of the nightmares and expansion, who has a Rogue Mega Evolution.
After catching Darkrai and quelling the threat to Lumiose, you can continue to survey distortions. By gathering enough data, you can learn to create special donuts that let you enter unique distortions containing Kyogre and Groudon, who revert to their primal forms. Then, Corbeau will tell Ansha to "wish really hard" for a distortion containing Rayquaza, which causes one to appear above Hotel Z. Doing more research allows you to enter this distortion and finally encounter Mega Rayquaza.
In the end, Ansha decides that she doesn't want to give Rayquaza to her mom just yet, since she still wants to stick around to make donuts and Diantha is still busy with her career anyway, so she lets the player look after it. Corbeau also begins offering Special Scans, which give the player the opportunity to find distortions containing Latios, Latias, Cobalion, Virizion, Terrakion, Meloetta, and Keldeo.
