Expectation value of an observable in quantum system
Roughly, the significance is that before measurement the system can be in a superposition of states, the wavefunction of which can be represented by a linear combination of the individual eigenfunctions

$\Psi = \sum_i c_i{\psi_i}$

However the very act of observing the system reduces the wavefunction to a single eigenstate due to interaction with the external world

For most cases, the expectation value is just the quantum mechanical way of calculating properties. Remember, the wavefunction is not observable! Only properties like density, dipole moment, etc are and the expectation value allows us a way to go from wavefunctions to properties via operators. 

These other answers may also be helpful 

https://physics.stackexchange.com/questions/67756/quantum-collapse
https://philosophy.stackexchange.com/questions/48425/is-collapse-of-the-quantum-state-conceptually-misnamed
https://physics.stackexchange.com/questions/151457/how-does-a-wavefunction-collapse