How can fields interaction give rise to particles? We say light a matter-wave, meaning along with its wave property it shows particle nature. But how can fields interaction (electric and magnetic) give rise to particles (photon)?
I wish someone could explain its answer in a simple graspable way.
 A: Field interaction does not "give rise to particles". In fact, field interaction makes it particularly difficult to speak of particles. 
To understand how particle states and fields are interrelated, we must employ quantum field theory. This answer of mine roughly sketches how a particle state is defined in QFT - and the fact is that such particle states are usually only defined in the limiting case where the fields are free, that is, non-interacting. To each "mode" of the field - each possible momentum, each possible frequency - there belongs a corresponding idealized particle.
But there is no "how" here. This is simply how quantum field theory defines particle states, and it turns out that this a) allows for predictions that match experiment amazingly well b) reduces in all meaningful classical limits to the correct classical case. One example for reproducing the correct classical behaviour is that the QFT notion of taking free particles, turning on an interaction, and shutting it off again to get particles again indeed reproduces the Coulomb force.
A: As you said waves have particle nature. The correct statement will be to say that the quantized fluctuations of the field can be visualized as particles. An intuitive example would be to imagine yourself in a pond with no ripples. In such a situation you will not feel any thing but if the pond has ripples then you would feel as if something is hitting you i.e momentum transfer is taking place via scattering process. You might as well quantize the waves and call them particles. When fields interact, field fluctuations take place which in the Quantum Theory are quantized and are viewed as particles.
A: In a simple word, particles are thought to be excitations in fields.
Particles are not interactions but we notice particles when they interact because we are capable of noticing the change in interaction.
(How I see is it's not the EM field that creates photon but photon itself is an excitation of the quantum electromagnetic field.)
A: In a simple graspable way? I always go back to the double-slit experiment and the many-worlds interpretation (though that is controversial, I admit).
The idea is that the particle is real, but it has taken every conceivable path to the target, each path in a separate world. These worlds have probability amplitude waves, so they can reinforce or cancel, with the result that some paths have high probability, and some have very low or vanishing probability. That's how you get the interference pattern. (I'm afraid this is not a very good explanation. Go back to Feynman, as I did.)
P.S. I did work in quantum computation some time back, and I found the many-worlds model the easiest way to think about it.
