What was Intel ME trying to solve that couldn't be done without it?
ME is an independent platform that runs parallel with the main CPU. ME has it's own CPU, memory, bus, etc. The general purpose is to provide an isolated subsystem on which to run security and management applications.
AMT's out-of-band remote access allows support to access the computer when the OS isn't or can't be loaded.
From the IT and security perspectives, these features are very valuable.
EDIT: If you want more info: