What is the cause of electric force?

Electric force is observed during the interaction of two charged objects. What is exactly happening to make this force possible?

From a Classical Mechanics prospective this question has no answer, in the sense that we don't prove the existence of the electric force, we just postulate it! We simply state that there is an electric force described by the Coulomb's law: $$\vec{F}=\frac{1}{4\pi \varepsilon _0}\frac{q_1q_2}{r^2}\hat{r}$$ and this assumption allows us to accurately predict a pletora of physical phenomena, so we assume that it is right! Things like this are the reson why physics is an experimental discipline. We cannot prove everything we say, it's impossible! We need to assume some basic laws about how the universe work and then use these basic laws (a.k.a postulates) as building blocks to derive all the other results!

Of course we could bring in the quantum concept of bosons carrying forces and all that, there is indeed some kind of deeper quantum explaination, but this is just moving the problem one step back! The existence of the electric interaction is not something you prove even in quantum mechanics, it is just observed experimentally and so it is postulate as base for our reasoning. Maybe one day we will discover some deeper theory of nature that will allow us to have other, more essential, postulate and prove the existence of the electric interaction; but at this point in time this is just speculation.

Other than this you seem confused by the concept of electric field and test charge:

Test charge has no such field,why would it exert a force on source charge.I do get how a source charge will exert force on test charge but according to Coulomb's law a force must be exerted by test charge on the source charge in opposite direction,I can't see how that will happen.

In Classical Mechanics force arise from the interaction between a field (electric field in this case) and a charged particle, not from the interaction of two electric fields; or at least this is the classical way to interpret the equations. Also the concept of test charge is a limit case: to feel the electric force the test charge must have a charge, so it must produce an electric field! You can state that the charge is infinitesimal, but this implies that the generated electric field is also infinitesimal, not $0$!
The usefulness of the concept of test charge is not that it hasn't an electric field, it is that its electric field is small, and so you can approximate it as not being there!

I think the farthest we can go on this is that the "cause" of electric force is basically that electric charge is a property that certain particles have which "sensitizes" or "hooks them in" to the electromagnetic field. Once "sensitized" thereto, they can act as a transmitter and receiver, of sorts, for energy and momentum from that field, such that they can put in or take out such therefrom. The amount of charge determines how strongly they are sensitized, how readily they give up or receive momentum, and the field acts as the mediator. In fact, they are both continuously acting as such, always both giving and receiving from the field.

Given two charged particles, when one particle puts energy and momentum in. Then it propagates through the field. Some of it gets to the other particle, which receives it, and changes its state of motion, and likewise it goes the other way. The further away the distant particle, the less momentum it receives from the source particle because more is lost bleeding into space, and conversely, so the force is weaker. The only trick here is that these processes are happening all the time in a continuous manner, and not in pulsating "episodes" of back-and-forth.

And by definition, this flow of momentum into and out of an object is what a "force" is.