For an event that can occur in many ways, why is the wavefunction of the event the sum of wavevfunction for each way separately? The wavefunction of identical particles is given as: $$\psi_{1,2} (x_1,x_2) = \psi_1(x_1)\psi_2(x_2) + \psi_2(x_1)\psi_1(x_2)$$ . Why is it so? Why is it the sum of the two states? What is the explanation behind this? Yes, I know that wavefunction being a linear combination of the solutions of Schrodinger's equation is a solution itself. But that does answer only mathematically. 
In order to get a deep insight, I again read Feynman's introductory lectures where he wrote: 

When an event can occur in several ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference. $$\phi = \phi_1 +\phi_2$$ .

Why is it so?
 A: You are making confusion between two different things: the first is why the wave function of two identical particles is what it is, the second is the probability of an event happening in several ways (which has nothing to do with physics but is just the definition of unions of probabilities).
As per the first question: given $\mathcal{H}_1$ and $\mathcal{H}_2$ as the Hilbert spaces of states for the solutions of the equations of motions for the particles $1$ and $2$, the Hilbert space of the system composed by both identical particles is the symmetric (or anti-symmetric) tensor product thereof, and therefore its states can be expressed as any symmetric (or anti-symmetric) linear combinations of the initial states $\psi_1,\psi_2$ of the two particles. This is the so called Pauli principle which is proven true experimentally, no other reason than that.
As per the second question: given $A$ and $B$ as two different events, the probability of either of the two is simply, by definition, $P(A \cup B) = P(A) + P(B)$. If you then interpret the wave function modulus square as a probability density function then Feynman's formula automatically follows.
