In the case of the failure the other week, put very simply it is this:
In normal operations, the computer passes flight details between sectors and centres automatically much quicker than a human can do. It is called electronic coordination.
When the system fails, this process has to be done by a human to another by phone. The receiving ATSA has to write down the details on a paper strip (or two, or three), give these to the controller, and then pass on the details to the next sector, where the process is repeated. Meantime, the phone is ringing from the previous sector with details on a new flight. That all takes a while and clogs up the system.
Is it safe? Absolutely
Does it work? Yes
Is it quick? No