Why are there more than one RF amplifier in a TRF receiver?
Radio receivers need to have very high gain (much higher that a single stage can provide) because the signal at the antenna may be very weak. The detector requires a strong signal to work properly, so a high gain is required before it. In a TRF receiver all amplification is done at the incoming signal frequency, so it needs multiple RF amplifiers. 

Superheterodyne receivers do most of their amplification at a fixed 'intermediate' frequency, which is usually lower than the incoming RF signal. This lower frequency is still RF, but the amplifiers are called 'IF' stages to differentiate them from the front-end 'RF' stage. Most Superhet receivers have at least two IF stages, and the mixer stage may also produce a small gain.  

A lower frequency is generally easier to amplify, so less stages may be required in a Superhet than in a TRF receiver. The lower frequency may also be easier to shield against unwanted feedback from the output, and a narrower bandwidth is easier to achieve because the tuned circuits don't need to be as 'sharp' (since the bandwidth is a greater proportion of the center frequency). Another advantage of a fixed IF is that its bandwidth doesn't change as the radio is tuned from low to high frequencies, and each stage can be preset to a slightly different frequency to make the passband flatter.     

The RF amp in a Superhet receiver only has to be sharp enough to reject the 'image' frequency (a separation of twice the IF frequency) and provide enough gain to swamp out noise in the mixer. Therefore one RF stage is usually sufficient (and cheap Superhets often don't have any).

Early TRF radios had a separate tuning knob for each stage. This was OK because there were few local radio stations, so listeners could leave their sets permanently tuned to one frequency. As more stations came on air it became a problem. Ganging the tuning capacitors of several stages works, but becomes unwieldy when many stages are required. As more RF stages are ganged together it becomes harder to get them all tracking accurately, and they must be well shielded from each other to prevent feedback.  Superhet receivers only need a maximum of 3 ganged capacitors to do the RF amp, mixer and local oscillator.