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.
Well we know that from the start people of Konoha hated Naruto and were afraid of him because of him being the nine-tails Jinchuriki. But when he met Sasuke he felt that Sasuke was also alone, just like him. That Sasuke didn't hate him and wasn't afraid of him. And in time he developed a bond with him, A bond of understanding. Naruto felt that Sasuke is just like him. In the Zabuza incident Sasuke was ready to sacrifice himself to save Naruto, this had a great effect on him. From that day he developed an inexplicable bond with him.
I guess accepting someone means really acknowledging that person and Naruto always acknowledged Sasuke when it comes to competing. Naruto never cared as to what others are doing. He was pin pointed at Sasuke as Sasuke was getting a lot of attention and Naruto always wanted to be acknowledged and known like Sasuke. Sasuke never acknowledged much people in his life but he always acknowledged Naruto more than anyone as he always competed with him all the time.Both were catching up with each other. Both wanted to become stronger than other.