I'm not a commercial pilot but would love if someone who is could post a plausible hypothesis how this would ground the whole fleet (short of a simple bureaucratic decision to shut down if everything isn't working exactly normally).
An important part of the flight plan is weather conditions along the route. There are obvious safety concerns, but weather conditions are also responsible for the route chosen, the amount of fuel onboard the plane, and passenger comfort. Without weather information, the flight plan cannot be filed and the plane cannot fly. Keep in mind that there are specific regulatory standards that must be met with any information included in the flight plan. A pilot cannot simply open up the weather app on their phone for instance. The weather info needs to come from an approved source.
I have no specific information about Southwest's current issue, but my guess is the third party service they use for weather in the flight plan is out of service. If they can't transmit this info to aircraft in the air or on the ground, then nobody gets to fly today. I'm also guessing that an incident like this will result in backup solutions being put into place, and that there is an IT or systems engineer somewhere who is trying to hide their "I told you so" expression.
Edit: an example flight plan can be found here: https://www.simbrief.com/system/guide.php#ofpsample. This is from a site for sim pilots, but the format of the brief is the same as what a commercial airline would file for a flight.
In order to dispatch a flight you need to ensure that the weather at the arrival airport meets certain minima at that point in the future (hence the need for forecasts), and so does that at suitable alternative airports, and enough fuel is carried to be able to divert (depending on a heap of other factors such as aircraft load, capacity, prevailing winds enroute, at destination, time of day and so forth).
Aircraft are then also required to check the AWIS/ATIS or the Tower (usually the most reliable source of weather at an airfield) to ensure that the weather at the destination is within their safe margins.
There's the FAA requirements, and then airlines and aircraft and airports then also have their own established minimum requirements that all need to be within the safe range.
You can certainly dispatch a flight with potential updates and diversions as they progress / get closer to the destination, but with an outage and no means to update the enroute aircraft - there's usually no easy safe or legal way to dispatch them.
I also agree that, much like when the Space Shuttle blew up, there were engineers somewhere who immediately suspected exactly how and why.
Also, the nice thing about wfh is the possibility of screaming about the stupidity of your organization more conveniently...
It also wasn't so long ago that scheduled domestic US airliners crashed on a fairly regular basis. That doesn't happen anymore because of advances in technology, this just being one little piece of that.
edit: s/FFA/FAA/