Is conservation of energy violated when two bodies move closer due to gravitational pull? Suppose 2 objects are placed not too far away. These objects start moving towards each other due to gravitational pull but there is no application of external force. So why do These objects move? Where does this energy come from?
 A: The answer to the question is in the question itself. You wrote:

Suppose 2 objects are placed not too far away from themselves. These objects start moving towards each other due to gravity.

Then asked: 

Where does the energy come from?

The energy comes from whoever placed the two objects apart. They had to do work to get them into that position. 
You could just as well ask "Suppose I roll an object up a hill. Then I let go and the object rolls back down. Where did the energy come from?" It came from you!
A: There is a potential energy associated with gravity known as gravitational potential energy. It is this potential energy which is lost to compensate for the gain in kinetic energy. And this loss is exactly equal to the gain in the kinetic energy of the two bodies. Thus the total mechanical energy of the system is always conserved and energy conservation is valid. Physics works!
A: 
Does gravity break physics?

No. Physics is the study of how reality works. The "description" or "modelling" of however everything technical around us behaves. Thus, reality can't break physics anymore than reality can break e.g. language.
Reality can be surprising (because we didn't expect it) or unintuitive (because we aren't used to it) etc. but it can't break physics, since physics is just a description of it.

These objects start moving towards each other due to gravity. But we didn't apply any force, neither did the objects lose mass.

We may not have applied any force. But the world is more than just us. Electric charges apply forces on each other so thunder storms happen, regardless of whether or not human beings exist. Why couldn't gravitational forces exist just as well regardless of human beings being there and applying it?
What we can apply and do has got nothing to do with what the world can apply and do. In short: We, human beings, have got nothing to do with how the world works.

why do These objects move? Where does the energy come from?

The energy is already in the system as gravitational potential energy. Just like elastic potential energy is stored in a spring. Release it, and the elastic potential energy will cause either spring-end to start moving - to gain kinetic energy. Similarly, release a gravitational system (drop a ball, or let two nearby planets move freely) and the two objects will start moving towards each other and increase their kinetic energy.
Whenever constant forces act, such as the elastic spring force, electric forces and also gravitational forces, then those forces can do some effort if released. We could call such effort that is just waiting to be released: "potential". And so, we think of this scenario as having stored potential energy. The kinetic energy that is "created" is simply converted from this potential energy that we started out with.
With this invention of the concept of potential energy, suddenly the energy conservation law seems to always hold true. So, this concept is very, very useful.
A: Well as Steeven answered, the force just exists there and is how the Universe behaves regardless of our intervention with the systems. But I believe a more intuitive answer to your question might be provided by General Relativity which eliminates this problem of the origin of gravitational force. According to Einstein's General Theory of Relativity, Gravity is not a force but just objects following their normal path through spacetime which is warped due to the presence of mass, energy, stress or momentum.

The presence of those two objects curve the spacetime around them as shown in the above image. Now even if initially, they appear to be stationary, it is only because they are stationary in space but they are moving through time (getting older). Hence the two objects are moving through spacetime, but since the spacetime around them is curved due to each other's mass they move along those curved paths called "Geodesics" and get attracted to each other producing the illusion of a gravitational force.
A: The energy( gravitational) was already there  precisely because there were two masses separated by a finate distance. 
A: I'll answer the question as asked:

Two objects at rest accelerate towards each other under gravity, where does the energy come from?

The correct answer is "Physics doesn't have a clue, but we've agreed a bunch of analogies to use depending on the situation."  Honestly, most people really hate this kind of answer, but it's the truth.
Newtonian answer
Assuming you're working in the context of Newtonian Physics, the answer others have provided, that the energy comes from whoever put the things there, is considered good enough.  More properly, logic in all physics that works within the constraints of Newtons laws of motion have two core assumptions that are, not exactly wrong, but not exactly right.


*

*All bodies originate motionless at the center of mass.

*Energy is conserved, but the details are not important as long as you're consistent.


The first assumption is there because every pair of objects have "gravitational potential" that implies that work was done to move out from the gravity well.  We can handwave that just fine if talking about throwing objects in the air because it just doesn't matter so long as we maintain a zero sum exercise.  The energy you get out is the energy originally put in.
Similarly, converting between types of energy is fine so long as we're consistent, nothing in classical physics actually cares all that much what the true nature of kinetic or chemical energy is, it just cares that the equations work and energy doesn't magically appear from no where.  Again, just maintain a zero sum.
Where this breaks down is when there isn't a zero sum.  That asteroid that just crashed in to the atmosphere didn't start off anywhere near the Earth.  The rocket we just launched out of the solar system isn't coming back to cash out the gravitational potential energy we gave it, so does that energy sit around somewhere?  The answers to these questions are the same: You have exceeded the scope of the model and the results are undefined.
If you had a model that involved every energy conversion since the Big Bang, was limited to acceptable Classical forces, and was consistent with reality, there would still not be an answer, since we'd need to define how the universe came to be to account for that energy.
Relativistic answer, orbital version
Mass distorts spacetime.  It's like how you're convinced you're walking through the forest in a straight line but some how find yourself back where you started an hour later.  What is moving, and in what path, is entirely subjective to the reference frame.  So on a technical level, the two things aren't actually pulled towards each other so no force is being applied.  It's kind of like how the expansion of space can make things accelerate away from everything else.
You want to know how we do collisions in this model?  We don't, as far I know.  Ask a Classical physicist once you're close enough to see what you're about to hit.
Relativistic answer, bridging to other systems version
Maybe the two objects are only moving from each other's frame of reference and it's only the worldlines that intersect.  No, that doesn't seem to explain the explosion.  Uh, something something tensor field, something vector something... ok look, we don't actually have a consistent explanation of what mass or gravity are, let alone how they work.  Pick the interpretation that seems to fit best and just resort to Classical Mechanics if nothing too extreme is happening.  If you're within the event horizon of a black hole then nothing can save you now.
There ain't no c in Newton's Laws.
Quantum mechanical answer
Gravity you say.  The door's over there, we've got way bigger problems right now, like how any of this stuff makes any kind of sense.  We're literally made of abstract equations, aren't we... reaches for the alcohol
My brain every time I try to get a straight answer on this mess
Look, when two properly defined bodies are attracted to each other, and get together at an intimate distance, it just gets really hot, they make a big mess of what's around them, and sometimes you end up with some new little bodies shortly afterward.  Sometimes the things are joined at the hip for the rest of their lives, sometimes they just get together for some fireworks and go their separate ways.  So long as you manage to control your re-entry, everything will be fine... and if you misjudge your target's personal space you might go down in a blaze of glory before your crushing end.
So don't worry too much about how the two fell for each other, just accept that sometimes the heavens move in mysterious ways, and remember to practice safe orbital insertion procedures so you don't end up aborting the launch and/or losing that really expensive satellite you spend so long working on.
Also, if you have a working theory for bridging Classical, Relativistic and Quantum gravity, prepare to be very rich/famous.
