The adverb genuinely describes something done in a way that's real and true. If your friend is genuinely thrilled about your plan to backpack across China next summer, she means it.
When you say something genuinely, you're honest. If you're genuinely worried about your friend getting hurt on her motorcycle, you'll beg her to wear a helmet. One theory about the origin of genuinely is that it comes from the Latin word for "knee," genu, referencing a custom of a father acknowledging his paternity by holding his baby on his knee.