That's no current will flow through the secondary until the instantaneous voltage across the winding exceeds the instantaneous voltage across the capacitor (plus losses in the rectifier(s))
If you imagine a completely discharged capacitor, current will flow from the secondary through the rectifiers into the capacitor as soon as the voltage across the secondary is high enough to forward bias the diodes. In a bridge rectifier this is approx 1.2V. So at the voltage rises from zero to 1.3V, there is about 0.1 of a volt greater than the capacitor voltage (0) and thus current starts to flow and the capacitor will charge up to 0.1V.
As the half cycle proceeds all the way up to 34 volts (24 x 1.414) the capacitor is charged by the transformer.
As the voltage starts to drop back down to zero, no current flows into the capacitor because its voltage is higher than the voltage supplied by the transformer.
Once you apply a load to the output, it causes the capacitor to discharge. The rate of this discharge (in terms of voltage) is determined by the current drawn and the size (in uF) of the capacitor.
If we assume that the load is able to discharge the capacitor about 1 volt before the secondary voltage again exceeds the capacitor voltage (approx 33V now) then current flows for a very brief period while the secondary voltage is between 33 and 34 volts. During this time (perhaps 5% of the time) the entire amount of charge used by the load needs to be replenished. Thus the secondary current is approximately 20 times the load current for this brief period.
This high current may cause several effects, resistive losses, larger losses in diodes, transformer core saturation, etc., which mean that the capacitors cannot actually be charged back up to 34 volts. Thus, under load the voltage drops, and the ripple increases.
The voltage is not determined by the rating of the capacitor. You use a 50V capacitor in this design because 34 volts is less than 50V and it provides a comfortable margin. You could certainly not use a 24 volt capacitor (it would probably explode in fairly short order) and a 35 volt capacitor is very marginal and I wouldn't recommend it.
Indeed you may find that the unloaded voltage from the power supply is higher than calculated. This may be due to a number of factors:
1) At low currents the Vf of the diodes can be very small.
2) the transformer may be rated for 24V at normal load, and may have a higher voltage unloaded.
3) your mains may be slightly higher than it's rated voltage
