Anyway, the heartbleed bug pretty much allows an attacker to probe a server that will end up revealing the private key. Once an attacker knows the private key, they can decrypt session keys that have been sent to the server, and thus decrypt all of the encrypted traffic that goes back and forth between the browser and the server.
Another bit of magic with public key encryption is the notion of “digital signature.” Your browser can create a mathematical challenge using the public key that only someone with knowledge of the private key can solve. This is part of how a website proves to a browser that it is what it says it is. If an attacker learns the private key of some website, then it can masquerade as that site.
All in all, the capture of a server’s private key is a bad thing, and that is what this bug enables.