Why a Transistor is considered to be an active device?
There are three(*) common ideal passive devices: resistors, inductors, and capacitors; the behavior of each establishes a precise ratio between voltage and either instantaneous current, the derivative of current with respect to time, and the integral of current over time.  The behavior of a typical real-world resistor, inductor, or capacitor may be very accurately modeled as a combination of resistors, inductors, and capacitors wired together in some fashion.

(*) Some people argue for the existence of a fourth type of passive component, a "memrister", whose behavior would establish a ratio between the integral of voltage over time and the integral of current over time, but unlike the other ideal passive components which are well-approximated by real-world counterparts that don't require an outside power source, no memrister has been constructed that can come close to achieving that. 

As for why transistors are considered active, it's because they're not resistors, inductors, or capacitors, and they do not establish a ratio among voltage and current, its derivative, or its integral