When you do something disdainfully, you do it with disapproval or contempt. You might look disdainfully at an airplane passenger who's yelling at his young son.
When you act disdainfully toward someone, you're expressing your lack of respect — either for the person himself, or what he's doing. If your grandmother disapproves of just about everyone she meets, she'll probably glance disdainfully at your friends when they show up in their loud car to pick you up. The Old French root of disdainfully is desdeignier, "scorn, refuse, or repudiate," from des, "do the opposite of," and deignier, "treat with respect."