I understand why you might want to really know what motivates people to accept you as a friend, but the problem with asking them to explain is that they have probably never consciously thought about it, and they may have found the quesiton a bit confusing. I find that people general don't consciously think about things that "just work" without much effort on their part. So if some people find it quite easy to form and maintain friendship, they never stop to think about WHY that process works for them. It is only those of us who struggle with how this process works who ever think about it.
As a personal example, I have come to realize in the last few years that it makes me very happy to feel that I am helping other people to achieve their own happiness. So I have become very willing to do things for my kids and some of their friends (like giving them rides to visit friends or go to activities), even when I know that it is inconvenient for me. So I choose to express my affection for my kids and their close friends in this way because I have come to realize that I derive a very tangible benefit from it - a sense of satisfaction that I have helped to make THEM happy.
But most people are not consciously aware of what benefits they derive from relationships, and therefore have difficulty explaining why they choose to be in those relationships. If they are being your friends because they feel you need them, you should realize that this also means that they derive happiness or satisfaction from being needed by you, so it is mutually beneficial.