Why does a wave actually diffract? I know that waves diffract around a slit and this is due to the Huygens-Fresnel principle. But I never understand this in an intuitive wave that why does a wave become a spherical wave front at the slit? Huygens principle gives all the math behind this but does not actually explain that why does a wave bend round the edges.
Some might say that this happens because this a property of a wave that it forms a spherical wave front and the final wave is a result of superposition of those wave fronts. I know that, but it would be great if anyone could come up with an intuitive approach to this and actually explain why waves diffract or why do they form a different wave front at the slit. This question might be somewhat similar to, why do waves bend around corners.

Image source: https://www.ph.utexas.edu/~coker2/index.files/Diffraction.gif
 A: The best intuition is the well-defined mathematics underlying the concept. The simplest equation for the wave is
$$\frac{\partial^2}{\partial t^2} h = c^2 \frac{\partial^2}{\partial x^2} h + c^2\frac{\partial^2}{\partial y^2} h $$
Here, $c$ is the speed of the waves ("fundamental" physicists would think about the speed of light as the most well-known example of the equation).
The second time derivative of the height of the water at a given place $(x,y,t)$ at time $t$ is equal to the Laplacian (the sum of second $x$-derivative and $y$-derivative) of the same height.
One aspect of the intuition is to know why this equation is right for a given physical system. For example, if water has "bumps" on it, the equation says that there is a force that tries to "flatten" these bumps. Such an equation may be defined from a mechanical model of the water as a continuum, or water as a collection of many atoms, and so on.
At the end, the fundamental laws we know – the Standard Model of particle physics, for example – contain some wave equations (e.g. the Klein-Gordon equation for the Higgs field) at the fundamental level (with some extra non-linear terms). In this context, these equations can't be derived from anything "deeper" (except for string theory which has its own wave equations in the fundamental equations, too – and they can't be derived from something deeper, and if they can, I must say "and so on").
Another aspect is why the wave equation above implies the Hyugens principle. It does. If you study how the function $h(x,y,t)$ changes if $t$ is changed to $t+dt$, one may see that the height at the given point is affected by the heights in the previous moment and in the whole infinitesimal vicinity of the given point. That's why the disturbances are propagating in all directions, whether these directions are around the corner or not.
You may imagine that the surface of the water is a grid or net of many people holding the hands of their horizontal neighbors, and holding the legs (with their legs, please be skillful) of their vertical neighbors. The wave equation says that whenever a human in the net feels that he's higher than the average of his 4 neighbors, he tries to lift the neighbors in the upward direction. So this rule makes the perturbations – bumps on the water or on any field – spread in both vertical and horizontal directions, and because the other directions are combinations that may be obtained by successive moves, the disturbances spread in all directions. Whether there is a wall or "corner" at some finite distance makes no impact on the fact that the signals are spreading in all directions.
A: I'll take a stab at a less scientific or mathematical approach to the problem.
You can think of water molecules as wanting to make the surface of the water as flat as possible. Seeing as any body of water will eventually become still (flat surface) if no outside forces work on it, it makes sense intuitively.
Of course water molecules can only feel the forces caused by nearby water molecules. So all they are really trying is making their local bit of water flat, which eventually flattens the entire surface of the body of water.
A last thing to keep in mind is that it will take a while for a molecule to change direction. If its neighbours are pulling it up, it can gain quite a bit of speed. When one of its neighbours then starts going down again, it will take a while before this molecule has lost its momentum. (the two neighbouring molecules may very well stop being neighbours since their speed will differ too much.)
So what does this mean for waves?
Well, let's imagine you pull one molecule up a bit. This molecule will consequently pull up its neighbours, which will in turn pull up their neighbours etc. However, these neighbours are also pulling the initial molecule down (and so is gravity) so while the neighbours gain upwards speed, this initial molecule gains downward speed, until it actually drops lower than its neighbours (which are still going up). At this point the initial molecule starts slowing down since its neighbours are now pulling it up. This process repeats, with the initial molecule alternatingly being lower and higher than its direct neighbours. Since these neighbours also influence their neighbours and so on, this creates a wave. Since there are statisticly just as many neighbours in any direction (and water molecules are extremely small) this spreads out at almost exactly the same speed in each direction, this makes a circle.
Now let's consider what a straight wave looks like. You have a long (or infinite) line of molecules that are at a maximum height and their neighbours are lower the further away they are from the intial row of molecules. Until we get far enough away, at wich point the pattern repeats itself. This shape also seems to move in a direction perpendicular to the line. If we assume molecules only move up and down, this can only mean the particles that are to the right of the wave (if the wave is moving right) are moving up and the particles to the left of the wave are moving down. You can easily see how this would result in each molecule moving up and down periodically, which would result in exactly the way waves behave. Since the wave is straight, the neighbours in the direction parallel to the wave must be at the same height and have the same speed as one another. So the wave can only propagate in a direction perpendicular to the wave.
So what happens when the wave hits the wall?
When the wave hits a wall, molecules don't have any neighbours to be pulled up or down by in that direction. This allows them to move a bit more freely, which results in the wave seeming to bounce back (I won't get in to this much further)
At the hole in the wall though, the molecules inside the hole will logicaly start moving up and down. In turn their neighbours will do the same. The neighbours in the direction parallel to the waves won't be at the same height as them though. (since the wave can't get through a solid wall.) So this situation ends up looking a lot like example with the one intially moving molecule, which resulted in circular waves. And that's exactly what will happen.
note:
I simplified the matter enormously, but I believes it sketches a more or less accurate idea of how simple waves work.
A: I think you are looking at the question in a slightly backwards way.  It would be better phrased as: Why is it possible to have plane waves? 
In physics all point sources, wave sources which are smaller than the wavelength, generate outgoing spherical waves.  As an example consider throwing a rock into a pond; the outgoing waves are emitted equally in all directions.  Generating a plane wave, such as you have at the input of your image, requires taking many of these point sources and exciting them coherently such that their individual spherical waves add up to form a plane wave travelling in one direction.  In the water example this can be done by moving a large flat surface back and forth which creates an infinite number of point sources along its surface. 
Diffraction is the opposite of this.  You've managed to generate a plane wave by coherently combining a bunch of point sources.  Now you block the plane wave except at a gap which is comparable to the wavelength, and in doing so you extract one of the spherical waves which was generating the plane wave.  
A: As Chris Mueller pointed out, plane waves can simply be regarded as caused by a given distribution of point sources.
As to why point sources should form spherical waves, you can maybe think of it as a consequence of the definition itself of a "point source". Indeed, when you say "point source" you are implicitly defining an object which has no privileged direction (at least not that is relevant on that particular circumstance), i.e. an object which is spherically symmetric.
Given this, imagine you want to describe the form of some kind of "field" that is generated by this source. This must have spherical symmetry, because otherwise you would contradict your very definition of a point source: you could distinguish a privileged direction of emission.
Of course you could now go on and ask why should elementary sources have spherical symmetry. The answer is that they generally are not. Infact, anisotropic radiation is more common, if you are looking close enough to be able to detect it. See for example here.
A: The wave propagates in the direction perpendicular to the surface of the wave front. The bit of the wave front that passes through the hole now has a surface on all sides, so it propagates in all directions.
Why does the wave propagate perpendicular to the wave front? Because waves are caused by restoring forces that smooth out differences - if you pull a bit of taut string up, it snaps back down to be closer to the adjacent bits of the string. The wave front is the surface along which the amplitude of the wave is equal, so there's nothing to smooth out and no forces in that direction. The forces propagating the wave are all perpendicular to the wave front, along the amplitude gradient.
A: lots of very complex answers
The simple answer is this.
A wave is a distribution of pressure caused by things being closer together than their natural . This could be interpreted as therefor having a higher energy or entropy which they seek to shed. As such this happens in all directions ... so a point wave will dissipate in a spherical fashion.
A moving wavefront will also do this but also combining with the motion of its wave.
So diffraction is actually the mechanism of propagation of waves. A linear wave front is a very special case.
A: Try to stick thin branch across door opening, so that branch sides leans to opening sides. Then push branch forward. It can't go as it is, because it's too long and does not fit into opening, right ? So what happens next is that branch will try to bend along push force direction, then it can go through opening. Similarly wave front will bend, because otherwise it will not fit onto the slit, and it can't go through "by parts", because wave is integral entity.
A: The roots of why Diffraction happens lies in "Heisenberg's Uncertainty Principle", which you can understand at the following two resources. 


*

*Heisenberg's Uncertainty Principle Explained by Veritasium. https://www.youtube.com/watch?v=a8FTr2qMutA

*Heisenberg's Uncertainty Principle in action! by Dr. Walter Lewin. https://www.youtube.com/watch?v=0FGo8mi-5w4
Following is a quick summary of Diffraction on the basis of Heisenberg's Uncertainty Principle. 

Heisenberg's uncertainty principle tells us that it is impossible to
  simultaneously measure the position and momentum of a particle with
  infinite precision. In our everyday lives we virtually never come up
  against this limit, hence why it seems peculiar. In this experiment a
  laser is shone through a narrow slit onto a screen. As the slit is
  made narrower, the spot on the screen also becomes narrower. But at a
  certain point, the spot starts becoming wider. This is because the
  photons of light have been so localised at the slit that their
  horizontal momentum must become less well defined in order to satisfy
  Heisenberg's uncertainty principle.

