How can a higher power circuit share the ground with a lower powered circuit?
Because of a basic rule of electricity, current only flows around circuit loops.

When you have two circuits, with no other connection between them, you can tie them together at ANY point to form a common reference voltage as shown below. No current will actually pass thought that connection without a second connection to complete a loop.







simulate this circuit &ndash; Schematic created using CircuitLab

Once you have created that common reference it is then possible to pass signal voltages between each side as shown here.







simulate this circuit

If you notice, the current driving the lamp only returns on the 60V side, while the base current returns on the Arduino side.

This is quite safe and normal. 

The issue with mixing grounds occcurs when you pass the returning high current  through the control sides low current circuits as shown below.





That creates noise and reference level changes on the control side which can cause unpredictable behaviour in the low voltage electronics.



The circuit in your image falls into the correct wiring category.

ADDITION FOR COMPLETENESS.

Note I said... "At any point". The following circuit is ALSO acceptable.







simulate this circuit