What you have shown us is a classic oscilloscope schematic. I designed 'scopes in the 1960 and 70s. Judging by the load resistors (15k) of Q18 & 19 this must be a very low bandwidth, low performance scope with low sweep speeds. These transistors form the output stage of the X amplifier which drive the plates of the CRT. CRTs require around 5 -10 volts of signal per division. So to deflect the spot across the screen (assuming it has 10cm wide screen) requires somewhere between 50 and 100 volts pk-pk. This signal appears at the collectors of 'push-pull' stage Q18 & 19, so between 25 and 50 volts per side in antiphase. The gain of the output stage is roughly (15k +15k)/500 ohms or 60 X when the gain potentiometer is mid-position.
Q16 & 17 are emitter followers. This is necessary to drive the reflected capacitance from the output stage. Q18 & 19 will have a collector to base capacitance (Cob) of somewhere between 5 & 10 pF. This is multiplied by the stage gain. This is called the Miller capacitance. Hence Q16 & 17 have to drive around 300 to 600 pF
You will see the Y amplifier uses a different technique. It uses a cascode stage consisting of Q11 & 13 and Q12 & 14. Again it is a push-pull output stage but because the output transistors are used in a grounded base circuit there is no Miller capacitance, hence this kind of technique lends itself to higher bandwidths.
Any more questions, just ask!