Promising things are encouraging or hopeful — they give you every sign that they'll be successful in the future. A promising early spring hints that the rest of the season will be warm and full of flowers.
A promising ballet dancer is young, but off to a very talented start. If you pick up a novel in a bookstore and the first few pages look promising, you'll probably buy it. Promising things are full of promise, "an indication that something will occur." The Latin root is promittere, "put forth," from pro-, "forward," and mittere, "send."