What is the purpose of Vsense terminal in some amplifiers?
This is a better data sheet for the LA6510 power op-amp and, it shows how Vsense should be used: -

Basically Vsense is a pin that can be used to sense the voltage developed across RSC. In other words, it's an output current limit protection facility.

I tried with 1k resistors wired as in application note but the output
seem to oscillate with high frequency (1MHz) noise of several volts.

The value of the resistor will be in the ohms range and not the kohm range: -

The data sheet I have linked gives values for components.