To embolden someone is to inspire them. You might embolden your brother to try out for a play by enthusiastically praising his singing and dancing abilities.
Shy kids sometimes need a friend or a teacher who will embolden them to speak up for themselves, and a good grade in a difficult class will embolden a student to attempt more challenging courses. When you cheer up a friend or encourage a coworker, you embolden those people. To embolden is "to make bold," from the Old English root word beald, which means "brave, confident, or strong."