This is an ADD ON, type. so what I figure is this.
For low beam I will explain (same for hi beam only the second part ofsystem)
Your stock wiring to 55W goes to the grey plug...(why didn't you post pic?? you may ask, the answer is I don't know how yet)
Grey plug will provide power to 4 pin relay (black device up front) and that relay will now be the power switch to your 80W lights.
Light switch on -> stock relay ON -> power goes through stock thin wires to stock plug-> Power to New Relay-> New Relay On.
then
New relay power from (+) source through heavier wire -> to Relay -> Out relay through heavier wire -> new plug -> 80W bulb.
Power to the 80W lights will be from +battery (or +alternator output) to relay to the 80W light, through the thicker wires in the add on harness. This way more current is available to the lamp (80W) and the wires are designed to handle it, thus no melting.
On my Cavalier I did the 80W low beams, and it caused my system to dim alot at stop lights, and then radio was not getting enough current

, so I did the DIY hyper goround, which helped a bit, then did the relay add-on like what you are planning on buying but that did not help.
Make sure that your lamps can take the heat!! or they may melt and

deform.

Cavalier had glass lamp and glass reflector!!! so it was heat resistant.
I doubt that the camry is glass, and I do not know what temperature can the housing, reflector and lense can handle.