Does it refer to the creature's dimensions or to the area that a creature controls in combat? And if a creature changes in size, does that inherently entail a change in the creature's physical dimensions or merely its area of combat control?
Size is a category
Size itself is neither physical dimension nor space controlled, but is a category. How the category translates into physical dimensions or mass is not exactly defined. How it translates into space controlled is exactly defined.
You can see so from the description of the Enlarge spell, that states:
The target’s size doubles in all dimensions, and its weight is multiplied by eight. This growth increases its size by one category — from Medium to Large, for example.
For characters, physical size is described on p. 17 PHB, under Size:
Characters of most races are Medium, a size category including creatures that are roughly 4 to 8 feet tall. Members of a few races are Small (between 2 and 4 feet tall), which means that certain rules of the game affect them differently.
From these examples in the rules it is clear that creatures become physically larger as they increase in size category.
The most common use of size is to determine how much space the creature controls in combat, and this space can also be used to determine object sizes. But in game mechanic terms, size influences many things, not all of which have to do with controlled space. Often how physical contests work between two creatures depends on their relative size, but there is more:
- What space the creature controls in combat (p. 191 PHB)
- Closely related to that, how many squares it takes up on a square grid or hexes on a hex grid battlemap (p. 251 DMG)
- Through how narrow a gap it can squeeze (p. 192 PHB)
- If it can move through an opposing creature's space (that is much larger or smaller, p. 191 PHB)
- How much food or water it needs per day (p. 111 DMG)
- How much load it can carry (p. 176 PHB)
- How easy it is to grapple the creature and how fast a creature can move when having grappled a creature of some other size (p. 195 PHB)
- If it can shove another creature (p. 195 PHB)
- If the creature can serve as a mount for another creature (p. 198 PHB)
- How spells (e.g. change self, alter self, dimension door, tiny hut, resilient sphere) can affect or transform a creature
- What size of weapons it uses (and how much damage they deal, p. 278 DMG)
- If it has disadvantage on using heavy weapons (p. 147 PHB)
- From how far away it can be seen underwater (p. 117 DMG)
- If magic items fit it (optionally, armor for certain races; oils, dimensional shackles)
- Optionally, how fast it is on initiative (p. 271 DMG)
- What hit dice it uses as a monster (p. 275 DMG)
This is a common approach in the 5e rules, similar to how creature type works: a label or category is attached to something, and then it can be used to interact with other game features.
RAW, its only mechanical effect is the space it controls in combat
Each creature takes up a different amount of space. The Size Categories table shows how much space a creature of a particular size controls in combat. [...]
There is an implication that bigger creatures are, well, bigger, but there is not an explicit rule.