Glycerin is a humectant, which means it binds water. Applied to skin it draws moisture from two places — the air, when the air is humid, and the deeper layers of your own skin — and holds it in the outer layer, where dryness is actually felt. Hydrated outer skin is more flexible, less prone to cracking, and less itchy. Your skin makes glycerol of its own and moves it around through channels called aquaporins, so this is not a foreign substance being imposed on it.
There is a little more to it than water. Glycerin helps the outer layer mature properly, supports the enzymes that let dead cells shed evenly rather than in flakes, and appears to help the barrier's lipid layers stay organised. In studies of dry and eczema-prone skin, glycerin-containing moisturizers improved scaling and roughness and were better tolerated than urea at comparable strengths, with less stinging. It does not treat any condition. It makes the skin more comfortable while something else does.
There is no group that should avoid glycerin. It is safe in pregnancy, while breastfeeding, on newborns, on broken skin, and on every skin tone. The only genuine gate is a patch-test-confirmed allergy to glycerin itself, which is rare enough that most dermatologists will not see a case in a career.
If your skin has ever felt worse after a hydrating serum, the likely explanation is dry air and a missing occlusive rather than a problem with glycerin. Put a moisturizer over it before deciding it does not suit you.
Immediately. Skin feels softer and less tight within minutes. This is a real effect and not just a cosmetic one — the water content of the outer layer genuinely rises.
Two to three days. Flaking and roughness ease, provided you are applying it to damp skin and sealing it in.
Two to four weeks. Studies in dry and eczema-prone skin show sustained improvement in scaling, roughness and hydration over about this period.
When you stop. Within a few days the skin returns to how it was. Glycerin holds water; it does not change how your skin behaves, so the benefit lasts exactly as long as you keep using it.