Why so friendly? They love to beg for food -- an apple, carrot, muffin, corn chips, whatever. They're not fussy.
The begging burros are one of the most popular attractions in Custer State Park. Children love feeding them. Parents do, too. If you search Google Images for Custer State Park, you're likely to see as many burro pictures and bison photos.
Most places discourage feeding wildlife, but the begging burros are an exception. For one thing, they are not wildlife in the strictest sense. They descended from pack animals once used in the park. I've heard they were used for treks to the summit of Harney Peak, At some point, the animals' owner simply let them go, and they've been roaming the park ever since.