West said that the pair had spoken privately about the incident and that Swift had explained how she had to stick up for West when people made accusations about him.
In an interview on New York radio station Hot 97, West admitted that his decision to storm the stage at the awards show was a "mistake" and that he had failed in his "responsibility" to his fans.
When asked what Swift had told him when the pair met, West initially joked: "She said, 'I just want to thank you so much,'" before adding: "No, that's not what she said. She just talked about how people would come up to her in the grocery store and bring it up, and she would defend me.
"It's an amazing, compelling situation. The situation is bigger to me than the Bush moment. It's bigger than a lot of things. I try to downplay it, but for the sake of being able to try to sell records and just get acceptance on that level that I did as a pop artist, there's just so few black men make it that far.
"That's a responsibility. That's why so many fans of mine were upset because they're like, 'Man, you've got a powerful situation where you can put your music out like that and do awards shows and everything. You can't be so reckless with your opinion. Like, we can agree with you but you've got to play it in another type of way, because you can't throw away the opportunity.'"
He added: "I made a mistake. My timing was definitely, extremely off, and the bigger plan, the bigger fight - how do you go about it? How do you go about getting it done? And the thing is, it was all in the plan. Beyoncé was set up to win the video of the year. If I hadn't said anything, everything would have just went smoothly."