Each commercial flight in the US is required to file a flight plan before takeoff. This flight plan is created and approved by an FAA licensed dispatcher, who then tracks the progress of the flight and develops alternative routing as necessary.

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.