Ever see or use a pantograph? This is a mechanical device that allows you to draw an enlarged copy on a piece of paper when you trace a small image. A vacuum tube (and also a transistor) are just electronic versions of this. A small signal to a control grid in between cathode (source of electrons) and anode (the output), makes a bigger copy (i.e., more voltage) of the original signal.
That's why they are called "amplifiers" -- they are amplifying a small signal, or again, making a bigger copy of a smaller signal.