How does a transistor act as an amplifier considering energy conservation?

The base current in a transistor controls the collector current. The energy comes from the power supply. It is not generated within the transistor.

The 'catch' is that a transistor only controls the flow of current; it does not itself generate power. The power would come from some other part of the circuit, perhaps from the electric company via a power supply or from a battery.

Now, one possible point of confusion is that transistors can be represented as equivalent circuits that contain a so-called "current source". This sounds like it could generate power, yes? Well, it turns out that a "source" can actually either generate or absorb power, depending on the relationship between the current through it and the voltage across it. The main thing a "source" does is fix the current (in the case of a current source) or the voltage (in the case of a voltage source) to a specific value. For example, a lithium-ion battery acts like a voltage source. If you connect a resistor across the battery terminals, the battery will supply current to hold the voltage more-or-less constant. However, if you connect an external power supply, the battery will start to charge, absorbing energy while trying to hold the voltage constant.

Now, there are several different 'models' or 'equivalent circuits' of different types of transistors, all of which use dependent sources in some way. The trick is that these models are only valid under particular operating conditions, and it turns out that there isn't any set of conditions under which a transistor will ever generate power. This isn't a trick of the mathematics, the reason for this is that there is nothing inside of a transistor that's capable of generating power; the only thing a transistor can do is generate a voltage drop to oppose the flow of current. It usually turns out that transistors end up dissipating a lot of power and end up needing to be mounted on large heat sinks.

A BJT is generally used as a power amplifier and the the power gained by the output signal comes from the DC power source that it uses.

Amplifying current on its own can be done without power amplification using a transformer but, if you want a power amp (i.e. the product of volts and amperes increased) then you need a power source.