Crystal Oscillator Power
The first problem with the circuit shown is that it shows a schmitt trigger inverter and that is incorrect; it should be a regular inverter. The next problem is that I don't believe the 74HC19 is a recognized part number so, going to the site you linked they specify a 4049 device and that is indeed a non-schmitt trigger device. So, assuming an inverter of the type normally used in these circuits...

Resistor R1 ensures that the inverter is forced into linear amplification mode i.e. R1 applies negative feedback and biases the input at the mid-point where gain is maximum. Then it's a simple case of a little bit of noise causing the output to start to wobble a bit and the rest is history; the oscillations build on the effects of noise.