I want to know how Lindeberg came up with the condition which is sufficient for CLT to hold ? What is the intuition behind such an expression ?
This is what Lindeberg says himself about his inspiration: J.W. Lindeberg, Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung (1922).
So it would seem that Lindeberg got his inspiration from Lyapunov. 


It was Lyapunov who introduced the method of characteristic functions and used it to derive CLT under certain conditions on moments of the random variables involved. His moment conditions were sufficient but not necessary for CLT. Lindeberg conditions are weakened Lyapunov conditions. Their advantage is that under additional assumptions they are necessary and sufficient for CLT. Lindeberg's conditions are tightly connected to "uniform smallness": the contribution of one concrete r.v. into the sum is small compared to the entire sum. A classical book on i.i.d. summation is by Gnedenko & Kolmogorov. A more recent, almost exhaustive and selfcontained monograph on classical summation theory is by V.V.Petrov. 

