Why do Russians say “с днём рождения” for happy birthday?

С днём рождения! is “happy birthday,” but word for word it says “with the day of birth.” The full sentence is поздравляю с днём рождения, “I congratulate you on your birthday”; Russians drop поздравляю and keep the с. The same frame gives every holiday greeting: с Новым годом!, с праздником!

After с the noun goes into the instrumental: день → днём, Новый год → Новым годом. Birthday wishes often go on with пусть..., “may...”: this one says пусть неудачи стороною обходят тебя, “may bad luck pass you by.”