I used to work on one of these systems.
The flight planning system takes inputs from several sources - weather forecasts, notices about airspace closures, etc. (NOTAMs), and booking info - and creates an optimal flight plan for the aircraft.
A modern airline doesn't have enough flight planning staff to take over manually if the system fails, so if your flight planning goes out, your fleet is gradually grounded.
The large number of servers is due to the optimization problem. You need to take into account the flight conditions and fuel costs in different locations in order to decide your route, altitude, and fuel loading. Since fuel is a huge percent of the operating cost of the airline, it pays to invest a little extra computing power into optimizing these and save a bit fuel on each flight.
Our system had lots of redundancy but, with all the data feeds, there are lots of moving parts. It's not hard to imagine a scenario where, for example, you get everything transferred over to your disaster recovery site, but for some reason the weather feed isn't coming in and you can't make flight plans.