Why is Boneh-Franklin IBE scheme secure? What should an adversary do to break it? Even after reading their paper I am still not able to understand why the scheme is secure and what an adversary should do to break it. This question might look stupid to most of you but I don't have mathematics background to delve this deep into cryptography.
The proposed scheme is IND-ID-CPA secure and is semantically secure in a random-oracle model.
From the paper:

The proof of security for our IBE system makes
use of a weaker notion of security known as semantic security (also known as semantic security against
a chosen plaintext attack). Semantic security is similar to chosen ciphertext security (IND-IDCCA) except that the adversary is more limited;

And detailing IND-ID-CPA:

We say that an identity-based encryption scheme $E$ is semantically
secure (IND-ID-CPA) if no polynomially bounded adversary $A$ has a non-negligible advantage against
the Challenger in the following IND-ID-CPA game:

The paper then goes into detail (which I suppose you've read).  But basically the game involves the adversary not being able to better than guess the value of an encrypted bit for a chosen public key.  The adversary has access as to as many other public/private keys as it wants.
The scheme is not IND-CCA secure.  As per this answer here one way to attack the scheme would be to gain access to a &quot;decryption oracle&quot;.