Why is a DC base voltage needed to attain faithful amplification?
Because of how transistors work. If you consider the NPN bipolar transistors for example (the ones on your diagram), they only amplify the input signal if the base is positively charged with respect to the emitter: that's when the isolation layer (called depletion layer) that spontaneously form between the base and the emitter is no longer enough to stop electrons from crossing and entering the base, forming the base current. If the collector is also biased positively with respect to the base (meaning, low positive voltage on the base, higher voltage on the collector), most of the electrons that will have crossed the gap will be sucked up into the collector, forming a higher collector current. Base current and collector current are in this way sort of proportional, you get amplification.

If you don't bias the base positively and just connect an AC source, half of the time the base will be negatively biased, strengthening the depletion layer and you'll get close to no current in the collector. The transistor is off, no amplification and in fact no current at all: that's why your output sinewave is clipped to zero for negative voltages.

Therefore you need to bias the base at half of the maximum voltage of your AC signal if you want to make sure the base never goes below conduction.


(source: learnabout-electronics.org)  

Above a random picture I picked from internet to illustrate what I'm saying. Apparently there is an animation on that page too, should you want to visualise it.

For completion, PNP bipolar transistors work the same way with different polarities. There are also different transistor families, the main "competitor" being field effect transistors.