Can someone explain what convention has been used to determine the
signs in the equation?
Sure. First note that the equation is for the loop clockwise (the signs would be reversed if counter-clockwise) that includes the voltage across $R_2$ ($v_1$) the voltage across the dependent voltage source, and the voltage across independent current source ($v_2$).
Now, before going 'round the loop, I would add a "plus" sign at the top of $R_2$ and at the top of current source. Why? This sign denotes the reference polarity, i.e., $v_1$ is positive when the top of $R_2$ is more positive than the bottom and similarly for the current source.
The rule for KVL is this: as you go around the loop, if you enter the "plus" marked end of the component, the voltage variable has a positive sign else it has a negative sign.
Starting at the circuit common node and going 'round the loop clockwise, we don't enter the "plus" end of $R_2$ so the voltage $v_1$ gets a negative sign. However, we do enter the "plus" end for both the voltage source and current source so those voltage variables get a plus sign.
That's really all there is to it.