What was the need for doing experiments to prove quantum entanglement?

A thought experiment is not really an experiment, but an idea. Science requires people to be able to test ideas with reproducible experiments. I can't reproduce a thought "experiment".

A reproducible physical experiment produces physical observations. See "Empirical research".

And you cannot communicate information through entanglement anyway. So my question again : why the experiments?

I think you will find that the technology is important in communication. Quantum cryptography is a way of sending secure messages with, for example, entangled photons used to send a key. If photons are intercepted between the sender and the receiver then the photons are lost and the information never reaches the receiver. The sender and receiver can tell something is amiss. If sender and receiver communicate successfully they can be sure that noone was able to listen in. The unique information passed between the sender and receiver can enables them to, for example share a private key, and communicate securely. The details can be read here and here.

So ... it is not possible to communicate by directly organizing a series of quantum states to be sent from one place to another, but it is possible to send information in the form of a random sequence of information from one place to another securely which can then used by sender and receiver to communicate securely.

I am not sure how much people had in mind quantum cryptography when they started these experiments, but it is an interesting spin-off that has come from this work.

(hope the wikipedia article linked above is helpful)

What you seem to not get from the paper is that the EPR-thought experiment actually made a prediction: It predicted that there are correlations within quantum systems that are stronger than in any possible classical system or any local hidden variable theory. The "spooky action at a distance" is just failing classical intuition. Don't read too much into it (I'll comment more below), let's first examine the stronger correlations:

This thought (that bipartite quantum states can exhibit stronger correlations than classically possible) is not really well presented in the EPR paper - and I believe that this is one reason, why experimentalists ignored it for decades. But other people, most prominently perhaps Bell, derived equations that hold for any classical system but do not hold for some entangled states - the easiest example being the CHSH-inequality. This is a testible quantity: You can produce states and test, whether they violate this inequality, if they do, that's a hallmark for a genuinely quantum phenomenon.

But why would you try to show this phenomenon over hundreds of kilometres? A few metres should be enough, shouldn't it? To show the existence of this phenomenon, a few metres would certainly be more than enough. The enterprise of producing entangled pairs over larger and larger distances that has only been tried recently and its due to the already linked to quantum teleportation protocol: While it is not possible to transmit an unknown quantum state via measurements and classical communication (i.e. phone calls), it is possible to transmit it by using entangled states. This opens possibilities for cryptography and information transmission, but for it to work, you'll need entangled states over long distances.

But what about spooky action at a distance and how does this not contradict the theory of relativity, which doesn't allow for instantaneous information transfer? The EPR paper was very much concerned with this "spooky action at a distance", but it is just a term stemming from classical intuition. Entangled states cannot transmit information faster than light (see multiple threads on this topic here, there is a mathematical account e.g. here: The choice of measurement basis on one half of an entangled state affects the other half. Can this be used to communicate faster than light?).

Even for things that seem very clear from the theory, you will want to check them. You asked

I mean, what were they expecting ----- were they expecting the states of the particles to be not in co-relation? How would they explain for that?

Well, of course they were expecting the entanglement. But finding that this is NOT there, would have been a huge thing - Quantum Mechanics needed to be amended!! As much sense as a theory might make, it must be subject to experimental verification in all aspects.

Similarly, most physicists were convinced for decades that the Higgs boson must be there and still we build ever larger experiments looking for it, since if we had NOT found the Higgs boson, we would have to re-think a large bit of what we know about particle physics.

EPR might seems a thought experiment, but it has been translated into an actual physic experiment, especially by physicist Alain Aspect.

You say quantum physic gives bizarre conclusion, that it is in an uncomfortable position, or that "philosophy of quantum mechanic" is at doubt. That's especially why it is important to really perform the "disturbing" experiments.

Many people are dissatisfied of quantum physic because it lacks an explanation compatible with common sense. This gives some idea like Broglie–Bohm theory, that are not very successful. They may hope that quantum theory will be replaced by another theory, a better one and one you can understand.

The thing is, outcomes of Alain Aspect experiment are facts. If another theory is to replace quantum mechanics, it must explain these facts. So this new theory will have some kind of weirdness.

In other words, these experiments shows that quantum theory is not bizarre : the world is bizarre. However, it is bizarre only if you want to stick with the common sense we inherited from our hunter-gatherer ancestors, we can understand it quite well with modern physics.

On the philosophical consequence of quantum physics, I have read and recommend this book by Bernard d'Espagnat "Veiled Reality: An Analysis of Quantum Mechanical Concepts". (I only regret this book doesn't really analyses Everett theory as well as other concepts.)

The experiments weren't tests of entanglement as such, but of Bell's inequality.

J. S. Bell proved, from assumptions that seemed simple and obviously correct, a result that was inconsistent with quantum mechanics. Many people believed that QM had to be correct and therefore that there had to be something wrong with the assumptions (though it wasn't clear what). Others thought it was plausible that the assumptions were correct and QM might actually be wrong. The arguments on both sides were good enough that it was worth settling the question experimentally.

I disagree with the accepted answer which basically just says "we should test everything". We can't test everything because we don't have infinite resources. The most basic criterion that any proposed experiment should satisfy before we fund it is that there be at least two different outcomes it could plausibly have. OP questioned the sense of doing a test of EPR on that basis, and I think they were right to do so. Before Bell's theorem, there really was no sense in a "test of entanglement" because no one (not even EPR) had come up with a reason to believe that the result of any particular experiment would be anything other than what QM predicted. EPR argued that QM was incomplete, but didn't know how a more complete theory would differ in its predictions.