One reason consensus problems are important is that they are very simple and they are kind of universal problems for distributed computing systems.
If we can solve consensus in an async distributed system we can use it to linearize actions on shared objects and obtain linearizability for shared objects.
For simplicity, how many problems can you think of which are simpler than agreeing on a value?
The impossibility result about consensus in (pure) async distributed systems tells us that we cannot solve problems we want to solve in (pure) async distributed systems without some additional "stuff". This leads to async models where we can solve consensus, e.g. randomized algorithms, fault detectors, partial synchrony models, etc.
This is also the reason why in practice algorithms that solve consensus like Lamport's Paxos, Google's Chubby, Apache ZooKeeper, and more recently Raft are at the core of distributed systems where we often want to replicate a state among servers.