Working of a Comparator based Oscillator Circuit
Definitions:


The inverting input of the opamp will be indicated by V-;
The non-inverting input of the opamp will be indicated by V+;


Analysis

Initial state:


The opamp has positive feedback through the two 10kΩ resistors and is configured as a comparator as a result. It's output will be either HIGH or LOW. When the opamp output is HIGH V+ = +1.5V and when the output is LOW V+ = - 1.5V.
A good start to analyze the circuit is to assume the capacitor is discharged. The voltage across the capacitor is 0V and therefore V- will be at 0V too;


Continuous cycle:


Let's asume the opamp output is HIGH, V+ = +1.5V. With V- = 0V, the capacitor will be charged through the 100Ω and with that V- slowly rises until V- reaches the same voltage as V+ = +1.5V. 
At that moment the opamp/comparator flips LOW and with that it pulls its own V+ input to -1.5V.
Now the capacitor starts to discharge slowly, down until the point where it reaches -1.5V. 
At that moment the opamp/comparator flips HIGH again, pulling its V+ input up to 1.5V and the capacitor is being charged again.
The cycle continues like that.