The core of the 'warm' analog sound vs 'cold' digital sound comes down to how digital works. Pic related is a rough approximation of analog (black) vs digital (red), though this example is greatly exaggerated and unless the music you're listening to was compressed to a dogshit bitrate it's not usually an audible difference. Some people (audiophiles) will insist that there's a huge difference. That's where you see terms like 'warm' 'bright' and 'full', but most of that is bullshit. What's really happening is that a majority of the time if someone is listening to vinyl and/or is an audiophile then they've probably also spent a lot more on their sound system, are actively paying more attention to the music, and have a bigger monetary investment in the experience (ie sunk cost fallacy "it has to sound better") or they purchased the record because it's already music they love. On top of this, most of the time if someone's listening to vinyl then it's usually older music that wasn't bogged down by the loudness war and also might have those trademark 'pops' and maybe background hiss just from the record itself being old and dusty. All that being said, I like getting records because it's nice to have a big piece of physical media for an album I like.

tl;dr there is a slight bit of truth in the sheer difference between how analog and digital waveforms work, but a majority of the 'warm analog' argument is bullshit and just comes down to conditions surrounding the listening experience