The honest answer is that a stylist has two hands free, a mirror behind you and years of practice. But most of the difference comes down to four things you can copy.
They dry your hair properly before styling
The most common home mistake is reaching for an iron on hair that's still slightly damp. Water inside the strand turns to steam under the plates, and that does real structural damage - the crackling sound is the giveaway. Rough-dry to about 90% before any heat tool touches your hair, and finish the last stretch with the dryer, not the iron.
They use tension
Smoothness comes from tension, not temperature. A round brush pulling the section taut as it dries is doing most of the work. Turning the heat up to compensate for a loose grip just damages hair without making it straighter.
They work in small sections
Sections that are too thick dry unevenly - the outside looks done while the inside is still damp, which is why the frizz reappears an hour later. Thinner sections, more passes, better result.
They finish cold
The cool shot on a dryer isn't a gimmick. Hair holds the shape it's in as it cools, so a blast of cold at the end of each section sets it. Skip it and the shape drops out within the hour.
What's worth buying
A heat protectant, first and always. After that, a decent dryer makes more difference than a decent iron - most of a good blow-dry happens before the iron comes out. Then a smoothing oil or serum on the ends only, applied to a finished style rather than worked through wet hair.