There are at least two likely reasons for this.
First, the freed slaves would not necessarily have trusted the federal government. There would have been little reason for them to assume that the federal government would care about them enough to have a Freedmen’s Bureau that would be completely effective. The federal government was, of course, run by whites, and the freed slaves would have been aware that even Northern whites might not really care about them very much. Therefore, it would have made sense to be a bit skeptical.
Second, the freed slaves would have wanted to be self-reliant. For centuries, African Americans had been enslaved and had been treated as less than fully human. One thing that they would have wanted was to prove that they were just as good as anyone else. The best way to do that is not to sit around and wait for someone else to take care of you. Instead, the best thing to do is to take care of yourself. Freed slaves would have seen self-reliance as an issue of pride. They would have wanted to show that they did not need other people to care for them.
Thus, the ex-slaves sought mutual aid both for pragmatic reasons and as a way of increasing their pride.