Athletes feel aches and pain from inflammation of their muscles.
Old persons frequently have inflammation in their joints which causes pain and stiffness.
Asthma sufferers have inflammation in their lungs.
Anyone (except hemophiliacs) who receives a skin cut has inflammation at the site of the cut.
It's an immune response and is a level of abstraction away from the actual observable symptoms, hence the general confusion about what it is.
So when people say 'eating sugar causes inflammation' what do they mean? How is THAT inflammation felt?
They do, in fact; a pro-inflammatory diet makes the threshold at which symptoms are felt from any of those other source lower, because inflammation has a substantial systemic component.
(It may produce unique symptoms of its own, too, but I think the main dietary concern with inflammation is systemic contribution to lowering thresholds to inflammatory symptoms when other triggers are present.)
We know there is inflammation there because the brains immune system goes apeshit, and we can see this under the microscope. Gliosis is the technical name.
High amounts of sugar that trigger the symptoms means >100g in 24 h or >60g per day over several days. The effect also depends on the food, probably because of how it is digested. Sugary water (fruit juice, soda) is worse than solid meals. Some forms of sugar are worse than others, e.g. I can eat some chocolate in moderation but others (e.g. Milka) I get a soar throat within minutes of eating a single piece (<10g).