What does Naruto mean when he said that Sasuke is the one that accepted him? Well, I've really been thinking for this for a long time now and still I can't understand what Naruto means when he said that Sasuke is the one that accepted him and his existence more than anyone.
The quote you are talking about,  most likely is this one used in Shippuuden


  Way back, I used to hate Sasuke... But once I got used to being with him I realized he's really a lot of fun... to be around...he's the person that, more than anyone else accepted me and my existence. Sasuke is my friend... and he represents bonds that I waited so long to receive, so that's why... - Uzumaki Naruto


By this point in the story you should already know how hated Naruto was in town, and how nobody would acknowledge him as he was the scary nine-tails. But then Sasuke came along, which in Naruto's eyes was similar to him as they where both 'alone'. They started to build up a rival like relationship, pretty much the same as Kakashi and Gai have a rivalship. At this point Sasuke did no longer see him as the "dangerous  9 tailed", and slowly started to acknowledge Naruto as an equal, a rival, and as a friend. 


   In episode 132 just after Sasuke says that skilled shinobi can read each others minds during combat, without even saying a word. There is a insight on how Naruto looks at Sasuke from 13:07 ~ 18:28 . Something similar happens in Shippuuden as well, just after the battle with Danzo.