How do transistors amplify weak signal?

A transistor controls the flow of current between the emitter and collector.

The larger current is supplied by the power supply, across the emitter and collector.

The flow is then controlled by a small current applied to the base connection.

So, a large current flow mimics the small signal current.

Limits are reached, when the transistor gets close to being fully switched on or off. This will cause distortion. To avoid this, more amplification is provided by a series of transistor amplifiers, one after the other.