You can turn on & off* LEDs with a transistor, but making a linear amplifier requires a few resistors & capacitors, and some careful design. Depending on how much gain and you need, you may need multiple stages (with at least one transistor in each stage). [u]Here[/u] is an example.
Sure, in the "old days" amplifiers were designed with discrete transistors but it's a LOT easier to use op-amps. With an [u]op-amp circuit[/u] you just need two resistors to set the gain. (But most op-amps only about enough current capability to drive one LED.)
- With PWM (built-into the Arduino) you are turning the voltage on & off rapidly and that allows you to dim an LED (or appear dim, or off, to the human eye). With PWM, you can use a single transistor or MOSFET in a simple circuit to "boost" the voltage/current and dim/control multiple LEDs.