어서 내려오다

come down, having done X

내려오다 is coming down toward the speaker, and the clause in front gives the cause or the manner. These lines cover coming down a mountain drenched in sweat, getting lost on the way down in excitement, leaving the stage out of embarrassment, and coming down to another city to introduce a gallery first. One line uses the handed-down sense, of something that has come down to us across thousands of years.