Question on Lenz law According to Lenz's law , the direction of current induced in a coil is such that it opposes the very cause which produce it
Why is this so?
 A: If we have a magnet moving into a solenoid, let's say with the North pole closest to the solenoid and the South pole further away.
The induced e.m.f in the solenoid turns it into an electromagnet with a North pole at one end and a South pole at the other.

If the direction of the induced current was such as to create a South at $A$, the magnet would be accelerated, gaining kinetic energy.  Energy is also generated as heat in the wires and this violates conservation of energy.
Therefore a North must be created at $A$ and a south at $B$
i.e. "the direction of the current induced in a coil is such that it opposes the cause that produce it".
A: One way to look at Lenz's Law is as an example of conservation of energy.  If a current is induced electrical energy is generated.  We don't get this energy for free.  It comes as the cost of doing mechanical work against the electromagnetic force which is opposing the inducing action.
A: Do you ask a technical question, solved by checking Faraday's law ?
Or a philosophical one, why is Faraday's law the way it is ?
If it were in the opposite way, a change of magnetic flux creating a current that increases it, any closed conducting loop that happens to undergo a change in the magnetic flux through it would induce a current that  increases it, indefinitely. That would be an infinite source of energy... The Universe could not survive with such laws....
A: 
Base image credits: WP base image
I will try to explain it as intuitive as possible using an analogy. The whole system behaves like a left handed screw (magnet $B_{1}$) and its corresponding right handed threads screw hole (current $I$ in a conductive loop, could be a spiral from a solenoid).
At case (a) nothing happens since we don't make any attempt to move the screw using our screwdriver in or out from the screw hole. Charges in the conductive loop wire are not pushed by the stationary magnet.
In case (b) the screw moves towards the hole by turning counterclockwise CCW (i.e. clockwise CW when the screw is observed from above its head, screw-in similar to right hand corkscrew rule) therefore the threads in the hole ($I$ current) move relative to the screw threads to the opposite direction thus clockwise CW when the hole is observed from bellow or counterclockwise CCW when observed from above. Similar to mechanical screw hole opposing the entrance of the screw, the current $I$ creates a magnetic field $B_{2}$ (see large orange vector at the inner side of the square black loop) that opposes the entrance of the magnet. As a reminder the arrow indicates the magnetic moment direction thus where the North magnetic pole is. Thus in case (b) here the N pole of the magnet is clashing with the N magnetic pole vector created inside the conductive black loop thus like poles and therefore we have repulsion. This is exactly the physical origin of the Lenz law expressed as the minus sign in the Faraday induction law:
$\varepsilon=-N \frac{\Delta \Phi}{\Delta t}$
\begin{array}{l}
\varepsilon=\text { induced voltage } \\
N=\text { number of loops } \\
\Delta \Phi=\text { change in magnetic flux } \\
\Delta t=\text { change in time }
\end{array}
In case (c) we now unscrew (i.e. magnet moves away from the loop) causing the opposite directional rotations of the field vectors from the ones described in case (b) above. Once more as in the mechanical analogy, the screw hole resists to the outward motion of the screw similar here the current $I$ creates a magnetic field inside the conductive black loop $B_{2}$ (see large orange vector) in the same direction of $B_{1}$ thus unlike poles are facing each other, N pole of magnet with S pole of conductive loop and therefore magnetic attraction is created that resists the outward motion of the magnet.
Cases (d) & (e) are similar to cases (b) and (c) but now the S pole of the magnet is  facing our conductive loop instead. Therefore, now our magnet behaves like a right handed screw (i.e. A Right handed screw, screws in by turning CCW on its head. A left handed screw, screws in by turning CW on its head) and our screw hole is now left handed relative to the screw. Again the magnetic field induced  in the conductive loop opposes the motion of the magnet.

The above described cases using the screw and screw hole analogy is actually close enough to reality since it is known that the magnetic field of a moving magnet induces a non-conservative rotational electrical field in a conductive loop wire that then creates current inside the loop which then creates a magnetic field around and along the loop (see orange field) that opposes the motion of the magnet moving towards or away from the loop's air cap, thus the Lenz law.
Here is a vector diagram of the electric field $E$, projected on a conductive loop generated by a moving magnet (i.e. colored circle at the center):

image credits: https://www.youtube.com/watch?v=OmlnGei1xo8
The total electric field $E$ shown above is irrotational in three dimensions and simply connected (i.e. has a vanishing curl, like a vortex) therefore is a conservative electric field in total. However, these vectors of the field which coincide spatially with the wire of the loop are forming a rotational non-conservative electric field induced inside the wire of the loop.
Notice also on the first top figure presented, that only the coherent direction vectors of the $B_{2}$ field which are inside the loop's air cap do matter in the Lenz effect. In a multiple loops coaxial configuration thus an electrical solenoid, these vectors from one spiral of the solenoid to other are coupled along the solenoid forming an almost homogeneous irrotational magnetic field $B_{2}$ inside the solenoid that opposes the motion of the magnet according to Lenz law.

