All the Apollo flights were observed by thousands of average citizens with normal telescopes in their backyards. The damage (O2 venting) to #13 was even observed. Yes the flights really happened.
EDIT: Laser reflectors were placed by the crews on the moon that still help mesure its distance today IIRC. Also sismographs placed by prevous missions detected the colision of #13's IVB stage on the moon.
