Check out the video on the Overview page if you can; it gives a few examples of the possibilities.
Just based on the information on the drop page, this is my understanding: ReSpeaker Core gives you an AI assistant like Google Assistant/Alexa/Siri, except you can program it to do whatever you want. When you use a product with one of those AI assistants, you're using a pre-programmed set of human commands (e.g. "Play Michael Jackson") and machine responses (plays Michael Jackson music using another app). You're working within the bounds of what Google/Amazon/Apple have designed. With ReSpeaker Core, you can make it do whatever you want. For example, when you say "Bye, Felicia", it could unlock your front door, open the garage door, tell you about the local road conditions, turn down your thermostat, and wait 2 minutes before turning off your lights. At least some of those things are possible with those other assistants, but only if you buy products that are all designed to work within that ecosystem.
ReSpeaker Core is also pretty small, allowing you to easily embed it within something you're making. In the video, it looks like they were often powering it from a small USB power bank.