How does a voltage varying from negative to positive applied to the grid of a triode amplify?
In a triode tube a high positive voltage in relationship to the filament is applied to the Plate, the filaments are a source of electrons. Without the grid there would be a large electron current from the filaments to the plate. The grid imposes a low negative potential between the plate and filaments. With a proper bias voltage, this effectively screens the filaments from the plate potential. Now by varying the voltage on the grid you can control the current flowing to the plate. Thus the voltage on the load of the plate will change in proportion to the voltage applied to the grid.
just a couple of little details. in most "modern" vacuum tubes, the electron source is electrically (but not thermally) isolated from the heating filaments and is called the "Cathode". a sorta corresponding alternate name for the Plate is "Anode". and the Grid is normally biased a little bit negative (with respect to the cathode, that's an important detail) because if the grid voltage is ever a little bit positive, it will act like a mini-plate and electron current will flow out of the grid terminal. but when the grid voltage is negative, since it is not heated, no current flows into it. it's a nice high-impedance input, like an op-amp. so to avoid that nonlinearity, the grid is biased negative and when a zero mean signal is applied to it, the instantaneous voltages varies around the bias point, but should never exceed 0 volts (w.r.t. cathode).
Separate names with a comma.