My experience is that you don't use them. They work poorly, which is why they are generally being replaced with newer more efficient dryers or removed altogether.
In theory, you can use them by pressing the button, then rubbing your hands together underneath in the airflow. If you're extremely patient and do this for long enough, you should eventually wind up with dry hands (not that anybody has ever waited this long). The dryer runs for such a long time precisely because it takes a long time to actually use one to dry your hands.
In practice, you use them by pressing the button, finding the air either too hot or too cold and trying to move your hands up and down so as to find hot air without burning yourself, realizing this will take forever, and either giving up or wiping your hands on your pants. More experienced users will typically short-cut the process and skip directly to the final step.
Don't just take my word for it though; complaining about these old hand dryers was a hot topic in the '90s, and earlier:
- Blowing off steam about those useless hot-air hand dryers in the restroom
- Electric Hand Dryer Takes Step to Blast Paper Towels: "The electric hand dryer is the public restroom's version of the Yugo: slow, ineffective and easily mocked."
- Dave Barry: "Electronic restroom hand dryers are miraculous labor-saving devices that work by shooting out a special kind of air, made from compressed sneezes, that actually makes your hands sticky without getting them dry."
- Gerald Nachman, The New York News (1975) - "In fact, just to avoid the offensive task of machine drying, people have given up washing altogether"
- What are Hyenas Laughing At, Anyway? - David Feldman (Google Books excerpt, full text not online)
You might also find interesting this Atlas Obscura article: The Weird History of Hand Dryers Will Blow You Away