What mechanism happens inside an inductor from the point when potential difference is applied?
There is a well known, well studied phenomenon known as electromagnetic induction that is mathematically described by Faraday' Law of Induction.

Essentially, if the current through a loop of wire is changing, the associated magnetic field threading the loop is changing and, by Faraday's Law, there is an associated induced emf.

Now, to your question:


  But what is happening in conductor so that the induced emf gets equal
  to E ? What is the picture of the mechanism ?


Let's assume the wire making up the loop is an ideal conductor or effectively so.  Now, recall that there can be no electric field inside an ideal conductor*.

With the voltage source E connected to the loop of wire, the only way for there to be no electric field in the conductor is for there to be an induced electric field that precisely cancels the field due to the voltage source.

Since the requirement for an induced emf is a time changing magnetic field and associated time changing current, the time rate of change of current is precisely the value required to induce an emf that cancels the applied voltage. 

*there are conditions in which this is not true but these are beyond the scope of this answer.