How does an active device work as an oscillator when its output is fed back to the input?
The statement is not complete on its own.  Overall, you need a gain greater than +1.  Over-unity power gain is a necessity, but it also has to be of the right phase, and the oscillator must NOT have over-unity gain at DC.

Less than unity gain means the signal is diminished each pass thru the amplifier, so any oscillation will die out.

Gain but inverted causes negative feedback when the output is fed back to the input.  This causes the system to stabilize and not oscillate.

Gain and in phase but DC coupled results in two stable states, each with the output driven to the two extremes.

It takes gain greater than 1, in phase, but DC gain less than 1 to make a oscillator.