Your diagram is a little off. Here's what it should look like:
The orange wire corresponds with the white wire in the video, and the brown wire corresponds with the red one. (I am talking specifically about the two wires connected to the motor; this correspondence might not hold for other wires.)
I'd like to explain what this is doing so that you understand what's going on. First, you should understand how a relay works:
When the relay is not energized (its normal, default state), the relay's COM (common) terminal is connected to its NC (normally closed) terminal, and NO (normally open) is isolated.
When the relay is energized, the COM terminal is connected to the NO terminal, and NC is isolated. On our relay carriers, the relay is energized when you put a high signal on the EN pin.
So when you connect the relay to the servo like this, the right motor terminal (green, COM) is connected to the left one (orange, NC) in the normal state, which brakes the motor. When you energize the relay, the right motor terminal becomes connected to the servo's internal control board (brown, NO) instead, allowing the servo to be operated as usual. Does that make sense?