In a Thermionic Triode, if the input signal is weak, where is the "strength" coming from that is amplifying the signal?

What actually causes the amplification within a vacuum tube? Is it the electrons being agitated/drawn to the anode from the cathode "amplifying" their energy from the time they leave the cathode to the time they hit the anode?

Is this the same as a ball bearing at rest being in range of an electromagnet, and then suddenly accelerating incredibly quickly as it is drawn to the magnet?