Vinegar contains acetic acid, a small, light molecule that partly evaporates even at room temperature. These vapors strongly irritate the smell receptors in the nose – hence the sharp odor.
This same acid is also why vinegar works so well against limescale stains: acetic acid reacts with limescale (calcium carbonate) and dissolves it, producing carbon dioxide among other things – you often see tiny bubbles form.
The stronger vinegar smells, the more acetic acid it contains – and usually the more effective it is against limescale. So the smell is essentially a direct byproduct of the cleaning power, not just an annoying side effect.