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Why a single phase motor is not a self starting motor

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Because single phase motor don't get required flux to revolve. That's why an auxiliary winding is required to produce required flux. That means at first you need to make it a two phase motor. Once it starts to revolve you can remove the auxiliary winding.
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For any machine or rotating electrical equipment, we require atleast two fluxes so that they could interact. But as the motor bears only one phase, two fluxes are not possible. Thus in order to obtain a second flux to be cut, we introduce phase shift (usually through condensers) of 90 degree(ideal). This new flux thus obtained interacts with the actual flux and then the motor starts to rotate.

If we need we could keep the capacitor connected for pf improvement or else the auxiliary winding could be removed by using centrifugal switches.
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For a motor to self start,there should exist phase shift between windings but now in single phase motors, flux produced oscillates on the same axis.This means that when you introduce another winding there will be an alternating flux.There will exist a rotation in the fluxes that will cause rotation of the rotor.The whole process is achieved by introduction of capacitors or auxillary winding or other methods.

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Reason of why single phase induction motors are not self-starting: When single phase AC supply is given to main winding it produces alternating flux. According to double field revolving theory, alternating flux can be represented by two opposite rotating flux of half magnitude. These oppositely rotating flux induce current in rotor & there interaction produces two opposite torque hence the net torque is Zero and the rotor remains standstill. Hence Single-phase induction motor is not self-starting.

OR

Single phase induction motor has distributed stator winding and a squirrel-cage rotor. When fed from a single-phase supply, its stator winding produces a flux (or field) which is only alternating i.e. one which alternates along one space axis only. It is not a synchronously revolving (or rotating) flux as in the case of a two or a three phase stator winding fed from a 2 of 3 phase supply. Now, alternating or pulsating flux acting on a stationary squirrel-cage rotor cannot produce rotation (only a revolving flux can produce rotation). That is why a single phase motor is not self-starting. 
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It needs phase shift to produce moving field with the main winding, so that's the advantage of the starting winding with capacitor in series connection,
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When single-phase AC supply is given to stator winding. It produces alternating flux.i.e. which alternates along one space axis only. It is not synchronously revolving (or rotating) flux, as in case of 3 phase stator winding, the fed cannot produce rotation. Hence single phase induction motor is not self-starting
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The motor mechanical degree 360 degree this must covered revolving field so single phase to starting need starting winding with 90 degree to main winding
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