Significance of Linearity in RF Power Amplifiers
By "straight line", they mean if you plotted a graph of "input amplitude" vs. "output amplitude", it would form a straight line. Distortion is a general term for when the output of an amplifier looks different to the signal coming in (if a sine wave went in and a square wave came out, we'd say that the signal had been greatly distorted).

Now, PA's don't specifically have to be linear, it depends on the application. For signals modulated in phase or frequency (FM radios), linearity isn't necessarily a big issue as the output amplitude is not that critical. However, non-linearity can produce harmonics at unwanted frequencies (intermodulation distortion is a good example) and is usually undesirable. 

Where linearity is a real issue is in high bandwidth systems with complex modulation schemes (WiFi, 4G, ADSL, cable TV - cable has VERY complex modulation schemes with up to a hundred amplitude levels or so). The reason is that the amplitude and phase carry information and if the amplifier is not very linear it can mess with the phases and the amplitude ratios (signals with amplitudes A and 2A might go in, but only A and 1.1A might come out). In these situations it's important that the signal coming out looks that same as the one coming in (albeit at a higher power). Otherwise, the decoder at the other end won't be able to tell the symbols apart which will lead to data corruption.

So do PA's have to be linear? Well, it depends...