Fuzzing is useful for finding bugs that may or may not be serious security issues. If a program crashes when you feed it invalid input, it could potentially crash in a way that allows code execution.

After finding a crash, a researcher will generally explore this to see if it allows for arbitrary code execution. If it does, it is possible for someone to create a weaponized exploit.

For a utility as widely-used as ffmpeg, any bugs pose a threat to many systems. Fuzzing it makes the Linux ecosystem safer for all of us (hopefully).

The alternative to crashing is exiting cleanly with an error. A crash can often be transformed into remote code execution with some work. That would mean that your computer could be taken over if you transcode the wrong video using ffmpeg. Which is pretty wrong.

There is nothing wrong with crashing if you can do it in a "safe" manner. If an invalid input allows an attacker to take over your computer, then that is a problem :). Especially for something as wide-spread as ffmpeg.

Video can often come from a less than perfectly reliable source (eg. satellite downlink), so video transcoders should be able to handle errors as gracefully as possible.

It's built into many products and workflows that accept user submitted content.