In quantum mechanics how can eigenfunction, eigenvalues, matrix methods give us values of real physical quantities? Eigenfunction, eigenvalues, eigenstates & matrix methods used in quantum mechanics seems purely mathematical.How can they give us values of real physical quantities in quantum mechanics?
 A: You don't need to go to QM to see that matrices are useful in representing physics; for example rotations - a very simple and physical motion - is represented by a matrix.
To see where the linear algebra in QM comes from, it might be useful to look at how it developed historically; for example, it was Einstein (according to Weinberg) that introduced the observable $A^n_m$ for the energy $E_n - E_m$ emitted in a transition of an electron from one orbital to another and used by Heisenberg in his Matrix Mechanics.
Though I'm not sure of the history, it wouldn't surprise me that this is why the spectrum of matrix - its set of eigenvalues - is called the spectrum.
A: The whole formalism of QM is based on a mathematical space called the "Hilbert space"  which is an infinite dimensional complex vector space equipped with an inner product.
A state is represented by a ray (and vectors $\psi$ belong to this ray) in this space. And the inner product of these states produces a probability to find a particle with the eigenvalue of an operator.
I.e.
$ O|\psi> = o|\psi> $
the object O is a hermetian operator which acts on the members of this vector space.
Indeed schrodinger wave mechanics formalism produces the same results but in general heisenbergs matrix mechanics are not only more illuminating but also more general and easier to work with in most cases.
For example, it is much simpler to study how a wavefunction $|\psi>$ evolves in space and time...especially the time evolution of a state is very simply a rotation of the state vector in Hilbert space
There's much much much more to the whole hilbert space formalism than this but this is merely to give you a flavor of what the whole fuss about maxtrix mechanics is
Ill add more later as my daughter is bugging me to play. :p
A: The entire point lies in the correspondence between the mathematical language as description of the universe and the events therein taking place. Physics aims to describe the quantitative behaviour of nature with the purpose of making predictions; in order to do so mankind has introduced a language to communicate and describe the above (mathematics). Such language is, of course, arbitrary and predictions must not depend on its choice.
Before going to quantum mechanics, let us deal with the easiest of the phenomena: point particle classical mechanics, namely Newton's laws. These relate external forces acting on a particle with its acceleration, which in turn is the second order derivative of the position (with respect to the time variable). The full description of a mechanical system is achieved solving a differential equation with initial conditions and this already sounds pretty complicated and mathematical to me, not to mention that in case of interacting particles we need at least tensor calculus on differentiable manifolds. So your question

How can they give us values of real physical quantities in quantum mechanics?

could become "How can tensor analysis describe the point particle and give rise to physical quantities?" and likewise for every other fundamental area of classical physics as well as thermodynamics, electromagnetism and so on and so forth. The answer is always the same: the mathematical description is nothing but a cooking recipe to get out some numbers, which happen to agree with the experimental results; whenever this happens we assume to have a good language and a good description of nature.
Coming back to your original question:

Eigenfunction, eigenvalues, eigenstates & matrix methods used in quantum mechanics seems purely mathematical.How can they give us values of real physical quantities in quantum mechanics?

they are not more mathematical than any other theory in physics. As we have found a recipe to describe the Newton's law, we have likewise found a recipe to describe quantum mechanics given in terms of operator algebras on Hilbert spaces and we accept it because it agrees with the experimental results at the end of the day. Had it not agreed, we would have chosen a different one (and by the way there is plenty of alternative descriptions of quantum mechanics).
