Why is the simulation required in these identity-based encryption security proofs? In identity-based encryption papers (i.e.this) it is very difficult for me to understand security proof. I don't know why they prove assumption (BDH, BDDH) for security of their scheme. Some uses random oracle and most of them try to simulate the scheme. Why is the simulation required in these security proofs?
(I assume you are asking about this paper based on your previous questions.)
They do not prove BDDH in their security proofs. 


  Assume that an attacker $A$ breaks IND-sMID-CPA of the above scheme
  with probability greater than $\epsilon$ within time $t$ [...]. We show that using $A$, one can construct an attacker $B$ for solving the
  BDDH problem.


In security proofs by reduction, we assume the proposition is not true (here: assume the proposed scheme is not IND-sMID-CPA) and show the proposition being false would imply some contradiction (here: solving BDDH problem which is believed to be hard). In this case assuming existence of attacker $A$ against IND-sMID-CPA security of the proposed scheme, they construct algorithm $B$ for solving the BDDH problem. Attacker $A$ is expected to participate in IND-sMID-CPA experiment, so through the reduction algorithm $B$ should simulate the experiment for $A$ and since this proof is in the random oracle model, $B$ should simulate random oracle as well.




  problem B is reducible to problem A if an algorithm for solving problem A efficiently (we call this algorithm $A$) (if it existed) could also be used as a subroutine to solve problem B efficiently.


In this context:

A = breaking IND-sMID-CPA
B = BDDH problem

reduction algorithm $B$ uses adversary $A$ as a subroutine. $B$ doesn't know how $A$ works; the only thing $B$ knows is that $A$ is expecting to attack problem A. so given an input instance of problem B (here BDDH), reduction algorithm should generate (simulate) an instance of problem A for $A$.

Chapter 3.3.2 of Introduction to Modern Cryptography explains "proofs by redcution".