LED Flash!
This is just a little more involved, requiring construction of a simple circuit and tapping into both the brake and running lights rather than one or the other (depending whether you installed the LED strip as an additional brake light or a running light)
The purpose of this circuit is to supply a lower voltage to the LED's (about 8) in "normal" running mode. A 330 ohm resistor is used to drop the voltage so the lights are a bit dimmer. When the brake is applied, the circuit allows a full 12 volts to reach the LED strip for full brightness.
As shown you will need a bit of wire about the size of the wire supplied with your LED strip (about 20 gauge.
2-4003 Diodes which provide a power gate so the 8 v doesn't make its way to the 12 volt brake light circuit and the 12 volt brake light power doesn't get into your running light circuit.
1-330 ohm resistor
Make sure the diode is downstream from the resistor on that "blue" leg.
All of this can take the space of a quarter. The parts and wiring needed is very small. Legs
of the "Y" need only be long enough to reach the appropriate wires under your seat.
By popular request, below is a photo of the actual circuit. Note the
whole thing is about an inch across (less if you want to tighten things up a
little more than I did here).