A drawstring waistband adds an internal or external cord that lets you fine-tune the fit beyond what elastic alone provides. This is especially handy if you're between sizes or have a body shape that standard sizing doesn't quite accommodate — you can snug things up without needing to size down and risk the shorts being too tight elsewhere.
Drawstring waistbands shine in mountain bike and commuter shorts, where you're moving around on the bike more and might need to re-adjust during a ride. They're also common in baggy styles, where the extra bulk of a drawstring is barely noticeable. The main downside is that a tightly cinched drawstring can create small pressure points on your midsection, and the hardware adds a bit of bulk compared to a simple elastic band.
If you're doing road cycling in a more aggressive position, you might prefer a wide comfort waistband or even bib shorts instead. But for trail riding, commuting, or casual spins where adjustability matters, a drawstring is a practical feature worth having.