Looking at figure two, the polarity has been marked such that, if you hooked it up to a battery like that, current would flow.
In figure one, the diode is reverse biased by the power supply, and current will not flow.
If you unplug the power supply, the solenoid drops out because the current stops flowing through the coil. However, the coil has inductance, which has the property of resisting any change in current flow - that is, the power supply will stop producing current, and the coil will start producing current in the same direction, until its magnetic field is totally gone.
Think of the stripe as an arrow showing you which way current is allowed to flow through it, and imagine you just unplugged the power supply. Understanding that current will no longer go through the power supply, you should be able to see that it will now flow out of the coil and through the diode.
If that diode were not installed, the coil can produce something called inductive kick, which can and will destroy the power supply by pushing voltage higher until it gets current flow - possibly through something that shouldn't be passing current.
You have the answer already and have asked for guidance ("But I don't know how we got there"), which is all I can give for homework questions...
Your question uses 'perfect' diodes, so no voltage drop and no leakage currents amongst other things.
Perfect diodes do not conduct when they are reverse-biased. Therefore any path that has a reverse-biased diode is open-circuit. When you're assessing the circuit, pretend that reverse-biased diode paths are not there.
Look at each diode in turn. Seeing the voltage it puts onto the resistor bottom-end, consider the effect on the other diodes.
This should take you most of the way to the answer. Then its just Ohm's Law at the end.
Best Answer
"BL" possibly means "back-light" and maybe it is a device that uses an anode and a cathode. Hence the net-name sounds fine to me and has zilch to do with D4020 and D4021.
Another hint is the "LCM" might stand for liquid crystal module i.e. a module containing a liquid crystal display AND a back light.