On the Proof Technique of Boneh-Lynn-Shacham Gap Signatures
The technique of "hopping" from one game to another, and proving the security through a "sequence of games" is not new. In particular, there are several papers which discuss these techniques, and exemplify them through various security proofs. The famous examples are:


Sequences of Games; A Tool for Taming Complexity in Security Proofs by Victor Shoup (2006)
Code-Based Game-Playing Proofs and the Security of Triple Encryption by Mihir Bellare and Phillip Rogaway (2008). Side note: This paper is a bit hard to grasp, so don't start with this one! The extended abstract of this paper is published in Eurocrypt 2006 under the name The Security of Triple Encryption and a Framework for Code-Based Game-Playing Proofs.
A Note on Game-Hopping Proofs by Alexander W. Dent (2007). Side note: This paper introduces another famous technique which was omitted from the above two papers.


Regarding your question, I use a quote from the last paper cited above:


  In a game hopping proof, we observe that an attacker running in a particular attack environment has an unknown probability of success. We then slowly alter the attack environment until the attacker’s success probability can be computed. We also bound the increase in the attacker’s success probability caused by the changes to the attack
  environment. Thus, we can deduce a bound for the attacker’s success probability
  in the original environment.


Thus, the reason for using several games is that in each game, we simplify the computation of some unknown probability, without altering it in a significant way. This continues until we reach a game in which the computation of the probability is easy enough.