Refraction and how light bends I have heard a particle nature explanation of how light continues to go with the same constant speed $c$ after it has passed through a denser medium. I also have come across how photon is absorbed by the dielectric molecules and then again re-emitted after a fleeting period of $10^{-15}$ seconds and that is how light is able to continue in its constant speed condition.
My questions are,

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*How does light bend at the interface of the two media? Could you please give an explanation without referring to the wave nature of light?(I know the Fermat's principle is just a vague truth)


*Also, how do light go in the same angle as it was in the rarer medium after it has passed through the denser medium?
 A: You have to realize that once one looks at the particle content of an electromagnetic wave, the theory that can describe the experiments is Quantum Field Theory (QFT), which is not for first year physics students.

In quantum field theory, quantum mechanical interactions between particles are described by interaction terms between the corresponding underlying quantum fields. These interactions are conveniently visualized by Feynman diagrams, that also serve as a formal tool to evaluate various processes.

Light in particular is described by  Quantum Electrodynamics (QED) . A light beam is composed by a superposition of an enormous number of photons each with energy $h\nu$, where $\nu$ is the frequency of the light beam and $h$ is Planck's constant. The classical electromagnetic field emerges from this confluence, not linearly but as described in this blog entry.
The classical wave mathematics is simpler in explaining the behavior of light waves, and as accurate.

How does light bend at the interface of the two media? Could you please give an explanation without referring to the wave nature of light?(I know the Fermat's principle is just a vague truth)

To go to the mathematics of this at the QED level is not an easy task. It is like asking "how do water waves reflect from the peer at the atomic level"  without using wave mechanics, just the atomic structure of water.For a QED description of  the behavior of light waves one trusts mathematics, which show how there exists a continuity between the QED formulation of light and the classical electromagnetic wave.

Also, how do light go in the same angle as it was in the rarer medium after it has passed through the denser medium?

It is the index of refraction of each medium that enters , and the two effects cancel. At the photon QED level it means that the probability of the photons to build up the light beam, the emergent beam, follows snell's law. A law in physics is the encapsulation of a lot of data/measurements and has the role of the axioms in mathematics. I.e. the laws of physics pick up the subset of the mathematical solutions that can fit, by construction of the theoretical models, the data. Snell's law applies to the classical behavior of light, but there is mathematical continuity between the single photon representation  and the zillions of photons in a beam.
A: Richard Feynman wrote a little book for the lay enthusiast simply called QED. In it he explains how light would always tend to follow particular paths, but not because the light is restricted to that particular path. According to Feynman, light takes all the possible path that there are. However, we only observe the light along those paths where it interferes constructively. (Oops, the wave nature again!) Unfortunately, it is not really possible to understand the behaviour of light if one tries to exclude the notion of phase. So even in quantum mechanics (or quantum field theory) where one tries to think of this in terms of little particles taking different paths, it comes down to the way they add up in terms of their phase differences that determines where they go.
So let's try to explain refraction ala Feynman. We have a situation where there is a light source on one side of the interface and a detector on the other side. On the opposite sides of the interface we have media with different refractive indices. The light source now sends out photons in all possible directions. At the interface these particles can again take all possible directions, but since we only detect those that go toward the detector, we only need to look at those that go in that direction. Now, Feynman associated with each photon a little `clock' or some devise that basically measures the phase for each photon along its path. The rate at with this clock advances is different in the two media. Now it happens that when the different photons arrive at the detectors, those with different settings on their clocks would tend to cancel each other. When we smoothly vary the paths that the photons took and we find that the clock changes drastically for those variations, then due to the cancellation the photons that took these paths do not contribute to the detection of photons by the detector. Instead, the path where a small variation in the path would not produce different readings on the clock is the path along which photons would contribute to the detection because it implies that the photons add constructively when the follow this path. It turns out this this prefered path is the one where one sees refraction.
Now one may think that this represents different physics from the physics associated with the classical description of light. However, nature always operates in the same way. It is just us humans that come up with different theories to try and look at something in a different way. Feynman understood the concept of Fermat's principle, which describes the variational nature of classical light and he then applied it to the notion of particles. In both cases one needs the idea of phase.
Additional clarifications
In view of some of the comments, perhaps we need to add some clarifying comments: 
The refraction of light is a linear process. It does not require the inclusion of interactions. It is simply the result of different dispersion relations in the two media that gives rise to this effect. One can in principle try to compute how the different dispersion relations come about by trying to calculate how photons are absorbed and re-emitted by the atoms in the medium, but that would a daunting task, not recommended when all one wants to know is how light is refractive at an interface. QED is an interacting quantum theory of the electromagnetic field and the fermions it couples to. One does not need QED to understand refraction, not even at the quantum level.
When we asks how one can explain refraction in terms of quantum theory, it is impportant to realize that the difference between the classical theory and the quantum theory in this context is artificial. In reality the two are exactly equivalent. When one considers a linear scenario (such as the refraction of light due to an interface) then the mathematical description for the classical field and that of a single photon are the same. 
So when we talk above of photons taking different paths, it is actually one and the same photon that takes these different paths simultaneously. The interference therefore takes place among the different paths (taken by the same photon) due to their difference in phase. This means that we don't need to consider multiple photons to understand this effect. In fact, multiple photons can bring other effects such as quantum entanglement that can give rise to quantum effects not seen in the classical scenario.
