These terms come from specific American subcultures, so understanding them requires understanding the subcultures they come from. "Dragging" someone is publicly calling someone out (identifying their flaws, insulting them). It is often used in a playful or hyperbolic way among friends. If a friend lightly criticizes another, a third observing friend might say "OOH DRAG HIM." This is popular among black women, gay men and drag queens. Roasting someone is a bit more broadly used, but it is used to denote that someone is being playfully but harshly criticized. A "roast" might refer to a comedy segment where a public figure is honored by being publicly ridiculed by comedians. Everyday internet users might feel "roasted" if they're made fun of or criticized on the internet.