What is the meaning of EM field having curl? We know that the "sourceful" fields like gravity or electric static field due to charge are all curl=0. But both E/M fields have curl and they mutually "curling" into each other, making them kind of "sourceful" in a very different way. What is the underlying source that makes EM field being able to curl? And if I have to guess, I would say it is spin.
 A: The reason is not really spin - we get circulating magnetic fields due to currents in straight wires, where spin is effectively absent.
Many would argue that the most fundamental quantity in electromagnetism is not the fields themselves, but what is called the electromagnetic vector potential, $\mathbf{A}$. This potential is directly 'sourced' from the charges and currents: it is (or can be made) straight when the currents are straight, and curled when the currents are curled and so on.
Now, the fields are then expressed as the curl of this potential (for simplicity, we are neglecting the electric field):
$$\mathbf{B} = \text{curl } \mathbf{A}$$
Now, we need some intuition for the $\text{curl}$. It gives something that goes "straight" when its input is "circulating" around (like a rapidly rotating turbine pulls in a lot of air). But $\mathbf{A}$ doesn't always circulate around. When it decays with distance (like from a current source), the $\text{curl}$ then produces something ($\mathbf{B}$) that goes around the source. (Now, the question of why the measurable $\mathbf{B}$ is the curl of some other quantity is perhaps best answered by the fact that "it works" - or demanding gauge invariance, whichever you prefer).
Now, what about electromagnetic waves? They are generated in part by the "curl-based" equations
$$\text{curl } \mathbf{B} = \frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$$
$$\text{curl } \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$
But the electric and magnetic fields don't have to curl around in them - it is just that if a vector field even varies a certain way perpendicular to itself in space, it has a $\text{curl}$ (this follows from the fundamental theorem of vector calculus - if it is not a pure gradient, it has a curl). One can typically have ordinary sinusoidal waves (propagating perpendicular to the fields) with no overall "circulation" in them. The sinusoidal variation itself implies nonzero $\text{curl}$s. Thus, the time-varying part of one field directly influences the space-varying part of the other field even when there is no curling around.
I guess the overall point here is that spin is not related to any of this. All of this follows from the classical theory, whereas spin is an essentially quantum mechanical property. You don't need something spinning to get those "curl" terms in Maxwell's equations.
A: We know that the "sourceful" fields like gravity or electric static field due to charge are all curl=0. But both E/M fields have curl and they mutually "curling" into each other, making them kind of "sourceful" in a very different way.
This gets tricky, but IMHO there is a way to understand it via a water analogy. See how AV23 said the most fundamental quantity in electromagnetism is not the fields themselves, but what is called the electromagnetic vector potential, A? Here's a depiction of it for a photon:

You'll be aware that the gravitational field is the derivative of potential? Take a look at electromagnetic radiation on Wikipedia and note this: “the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time”. Now sketch out the spatial derivative of the "hump" above. Plot the gradient. What you get is this:

That's your electric sinusoidal waveform. Notice how the Wiki article mentions the same phase? If you drew the time derivative of the hump, you get the same sine wave. Sounds confusing? Not if you imagine the hump is a troughless oceanic swell wave and you're in a canoe. As the wave passes you by, your canoe tilts. The tilt angle of your canoe denotes E, and the speed at which the tilt is changing denotes B. There aren't really two different fields, they're two aspects of "the greater whole". 
What is the underlying source that makes EM field being able to curl? And if I have to guess, I would say it is spin.
I'd say you're kind of right. Another word for curl is rot, which is short for rotor. It's to do with rotation. Your canoe starts off horizontal like this _, it rotates to this angle \, then it rotates back to horizontal _ again at the top of the hump, then it rotates to this angle /, then back to horizontal again once the wave has passed. So like AV23 was saying you have ordinary sinusoidal waves with no overall circulation, but they have a nonzero curl. Note that the current here is displacement current. Water surges up lifting you up, then back down again. Moving on from this to charged particles starts stretching the analogy, but see the picture of a spinor on Wikipedia? It's like the oceanic swell wave going round and round in circles. And like AV23 said, you don't need this spin to have curl terms.   
A: The notion that there is a curl involved is vector calculus voodoo.  You can write Maxwell's equations in special relativity, and there is no curl involved.
But either way, there is only the EM field and its source, which is current. Spin has nothing to do with it.
A: 
And if I have to guess, I would say it is spin.

You guess right. Electric field comes from electric charges from protons and electrons. Magnetic field comes from magnetic dipole moment from protons, electrons and neutrons too. And the magnetic dipole moment is connected 1 to 1 to the intrinsic spin.
