How do we know quantum entanglement exists/happens? We know that this strange behavior is there but I am not to able grasp, how do we know it exists? Have we ever tried any experiments confirming this? Do we make use of this?
 A: As with any phenomenon occuring in nature, we know it because we can measure it (well, actually what we can measure are phenomena like quantum nonlocality and quantum steering, but both are only possible due to quantum entanglement, and therefore those measurements prove that there is quantum entanglement).
The oldest example is the violation of Bell's inequality. Bell's inequality is an equality about expectation values of observables which holds for all separable (that is, unentangled) states, yet we can measure that real systems do violate Bell's inequality. States which violate Bell's inequality are called non-local because they are exactly the states which cannot be modelled with a local hidden variable theory (that's actually the original purpose of Bell's inequality).
Another simple example of an effect only possible with entanglement is dense coding (which also has been demonstrated experimentally). Imagine Alice wants to send Bob a message composed of bits. However, all she can send to Bob are systems which, when you measure them in any way, only give one of two results, and after that measurement, any further information which might have been in the sent system is gone. Now it seems obvious that you can send at most one bit using such a system. However it turns out that if this is a quantum system (a qubit), and actually part of a specific two-qubit state (called Bell state) where bob already has the other qubit (that is, the transmission of that qubit happened long before Alice had the bits she wants to send; indeed, it might even be that Bob created the pair and sent one of the Qubits to Alice), then Alice can encode two bits in this single qubit and send it to Bob, and then Bob can reliably read off both bits with the help of his part of the Bell state qubit pair. Again, that this can be experimentally demonstrated proves that entanglement exists.
