Why is the simulation required in these identity-based encryption 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".