Great question, and excellent answer, Joss.
In nutshell, blockchains (really Distributed Ledger Technology) allow us to create blueprints for customized economies. A successful economy has a set of actors, rules to follow, a determined way of exchanging value among these actors, and protections against fraud, seizures and mal-acting. This is the main reason why this technology is foundational, and will change the course of our lives in the 21st century. Payments is only the first use case (similar to how email was for the consumer internet), but we are only scratching the surface. Tokenization of real world assets will expand trade and exchange of value at a level not seen before in history.
PS: I write about similar topics at my blog: ash.money