I think a general idea behind integrating factor is the use of product rule. Since a general form is u(t)dy/dx +u(t)p(t)y = u(t)g(t), and we can observe a pattern of u'y+uy' on LHS. Thus the LHS becomes d(u(t)y)/dt, and it makes the equation easier to solve. More details are expanded in the textbook starting from P24 and those illustrations and examples are quite useful. I don't know if I have addressed your problem, if I did not make my point clear please comment below.
