Power supply PCB

As others said, it's mostly because it was easier to manufacture than solid layers for various reasons.

They also can be used in certain situations where you need controlled impedance on a very thin board. The traces width needed to get 'normal' impedances on such a thin board would be ridiculously narrow but the cross hatching changes the impedance characteristics on adjacent layers to allow wider traces for a given impedance.

If for some reason you need to do this, you can only route controlled impedance traces at 45 deg to the hatch pattern. This approach greatly increases mutual inductance between signals and consequently, cross-talk. Also note that this only works when the size of the hatch is much less than the length of the signal's rise time, this normally correlates to the frequency of the digital signals in question. As such, as frequency increases you reach a point where the hatch pattern would have to be so tightly spaced that you lose any benefit vs a solid plane.

In summary: Never use a cross hatched ground plane, unless you're stuck in some really weird situation. Modern PCB construction and assembly techniques no longer require it.

If you have traces forking off the main trace, or separate traces running directly to the regulator then these only need to be wide enough to handle the 500mA for the Arduino.

You can think of the power distribution a bit like a river with tributaries. The main "river" needs to handle the total power for everything and the tributaries need to handle the power for whatever they lead to. Remember to think about the full loop (e.g power and ground traces) for everything.
If the Arduino was directly in line to the servos (i.e. the servos were downstream of the Arduino, so it goes Regulator->Arduino->Servos) then the traces would need to be rated to handle the Servo current too.

Usually it is best to keep the high current stuff on a separate trace, or nearer to the supply, as the more of the route they share with sensitive components the more they can interfere. With planes a similar logic applies, though positioning is more important as you have to assume the path of least impedance will be taken (not always the shortest path)
Edit - As Kortuk mentions a plane is preferable if possible. A dedicated solid layer (i.e. no routing, or if some is necessary try and make it in the same direction to current flow so it does not have to "take the long way round") is best but a fill is certainly useful also.