Many colleges today receive teacher recommendations electronically, directly from the teachers who write them. But, at some high schools, teachers still snail-mail their references to colleges. In these cases, I always advise the student to give the teacher stamped envelopes addressed to the appropriate admission offices.
Thus, similarly, I recommend that you stamp an envelope, address it to the admission office, and mail this teacher’s reference to the college. (If there is more than one college involved, make copies of the reference and send one to each school.) Unlike in my own high school era many decades ago when references were always confidential, some teachers these days show students what they wrote … even when the teacher submitted the recommendation directly to colleges. Thus, the fact that you read your recommendation isn’t rare, and the teacher obviously intended for you to see it or he wouldn’t have handed it to you.
You didn’t follow the proper protocol but you didn’t do anything unethical either, so just mail the reference yourself. But, if you plan to ask this teacher—or any others—to write on your behalf in the future, and if these teachers will not be submitting their letters online, then present them with stamped envelopes so that you won’t be stuck in the middle again.