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

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.

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?).

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".