Analogue signals often have noise or other factors you want to get rid of. Maybe there's a low frequency drift you want out, or a high frequency noise you want out. From there, it goes to enhancing and more generally modifying the signal, as in edge detection, but the math is generally the same and so is the name "filtering." Or in position estimation from a signal, like a kalman filter or particle filter.

At that point filters are a hugely broad thing, and if you want to smooth an image, you might filter out the high frequency components (instant automatic airbrush) or maybe you want to remove all the blue (like a lens filter you might physically put in front of your camera) or even enhance all the blue. Then it's a small step to keeping the filtering name (yet again) for all sorts of signal/image/video auto-manipulations.

You start with wanting to get rid of noise or get rid of blue light and end up turning people's videos into cats. That's language for ya.

edit: I see other comments point to instagram. I think that is giving insta way to much credit. This is filtering as in signal processing. It's a common term in audio, image, and video processing.

https://en.wikipedia.org/wiki/Filter_(signal_processing)

https://en.wikipedia.org/wiki/Filter_(software)