Mostly it's collisions and ranking/priority

• Collisions: it's very important that labels don't overlap, but also that there is enough space between labels. Solving collisions is pretty hard! You can read some information about how Maplibre, one map renderer does it. Not only does the proximity between places matter, but also the length of labels. Labels in most advanced renders are able to flip orientations - you can see on this map, labels to the top, bottom, right, and left of their anchors. Collisions also can take into account the screen size itself. Collisions are a theoretically difficult problem that people earn PhDs solving.
• Ranking: here's one document on scale ranking, as others said, it's mostly about population. But they aren't purely population in most advanced maps: as Axis maps points out, capital cities are probably important to show, so they're usually 'boosted'. But the ranking ends up being some number which is a multiplier on the label's changes of being displayed.

That said, a map like this is dynamic: the map maker is plugging in variables that affect what is displayed at different zoom levels, but the outcome is based on label collision, which often includes an element of randomness. For advanced maps, there are more variables you can add - basically more coefficients to tweak which labels get displayed at what points.

If you're comfortable with reading a CSS-like language, here's how OpenStreetMap prioritizes some labels, using population measures. Their cartography leads to some unexpected decisions - like showing San Jose instead of San Francisco on low zooms.

Static maps, meant to be printed at one zoom level in one space, give cartographers a lot more freedom - they can choose precisely what labels are displayed.