Local-first software is a new type of multiplayer app (or game) that lives on your device, not the cloud.
Social networking and cloud apps have made it easy to connect and share with others over the internet. In return, you're required to hand over your data and identity, and follow their rules and requirements.
Back in the '90s, software worked whether you were online or offline, didn't sell your data or rent it back to you, and let you use it however you wanted.
Local-first software is a way to bring back ownership of your tools, and control of your own data.
For a more in-depth explanation, check out this article by Ink & Switch.