Their definition of the service is:
"A decentralized network based on cryptographic keypairs and that is not peer-to-peer, it is super simple and scalable and therefore has a chance of working."
You can read more here:
https://github.com/nostr-protocol/nostr
The short of it is that it's a decentralized version of Twitter. While this isn't exactly accurate, it's a good quick explanation of why people are using their twitter accounts to promote their nostr identity. I haven't used it personally, as I enjoy Twitter, but I can see how evolving a project like this could turn into a trusted and decentralized social network in the future. It feels very "web3" to me, which is what I find fascinating about those who are promoting it, as they have typically been anti-web3 in the past.