A lot of us grew up using lever guns, and we got used to seeing game animals over iron sights. My dad is left handed, and lefty bolt guns were outside his spending limits, so he shot lever guns. That means I grew up shooting them, including my great-grandfather's 30-30. Family traditions have plenty to do with how we view shooting and hunting.
Lever guns are also all-American type rifles, so there's a lot of national history and pride when you're looking at a lever action. Traditional lever guns have a solid, dependable, yet classy mystique.
Light weight is another big plus. Perfectly good, dependable lever guns have been available at 6 pounds or less for a long time, long before ultra-light bolt guns became all the rage. Combined with the easy, quick handling properties of most lever actions' stocks, that makes for a very handy hunting gun, especially at the close ranges where most game falls.
Price has always been a big plus, too. The basic Winchester '94 or Marlin 336 won't break the bank, and they're ready to use as soon as you've cleaned off the packing grease and sighted in with your hunting loads. I prefer a peep sight on my lever guns, as weight and bulk remain unchanged, the sight only costs about $35, and it offers "minute of venison" accuracy out to sane hunting ranges.
Lever actions appeal to traditionalists, cheap skates, sticks-in-the-mud, history buffs, brush hunters, walk-in hunters, and horse back hunters. In 444, 45-70, 405, 450 Marlin, and other heavy calibers, they appeal to folks that hunt big game that's kissing close. They also appeal to hunters that don't fall for the "latest and greatest" rifles and wonder calibers being sold to us by the gun magazines.
Lever action users know that you still need to be a good hunter, first, and a 180 grain spitzer bullet going 3500 fps out of a $900 bolt gun won't make you a better hunter, even with a scope the size of a pringles can on top of that rifle.
That slick working lever, and the stock staying glued in place while you operate it, are just the cherries on top.