Are We Lost's reply has part of the answer, but it doesn't explain the presence of the relay click you are hearing, so I think a more complete explanation is warranted...

Modern cars send control signals along the same wires that are used to actually illuminate it's lights. They are to detect whether the globes are blown or not by detecting the correct continuity and resistance on the circuit, and alerting the driver to a blown globe at the dash if they don't get the expected results. These signals, sometimes referred to as a "cold check" are of sufficiently short duration and power to check the circuit without actually illuminating the globe itself, so normally we're not even aware of them happening.

When you attach a trailer the car does the same thing on the trailer indicator circuits, but in this case open circuit doesn't indicate a blown globe, but that there's no trailer attached, whereas continuity and a certain level of resistance indicates there is one attached. It uses this knowledge to automatically turn on it's various 'smart towing features' if it detects a trailer is attached. These do things like de-activating the rear fog lights and reversing sensors on the car to prevent them being a nuisance when there's a trailer behind, and alter the car's  auto transmission and braking mapping to cope better with towing a heavy load and use the car's ABS circuit to automatically correct trailer sway.

So far so good, but there's two potential issues with this scenario when the trailer lights are LED.

First, LED lights don't create as much resistance as filament globes so often aren't detected by the car, so if the car doesn't know a trailer is attached it won't activate the extra towing features. Now most of these are nice to have convenience and safety features, but aren't essential, and the car will still tow the trailer and the trailer lights operate as normal. But, there's an easy solution as Are We Lost said, put load resistors in parallel with the LED indicators in the trailer to create enough resistance in the circuit for the car to detect it's there and activate the towing features.

Second is that the control pulses though not of sufficient duration or power to illuminate a traditional filament globe, or even an LED unit specifically designed to work on that car, can make the simple LED light units on a trailer flicker annoyingly. The solution to this one is to have the trailer indicator circuit operate a relay which in turn operates the trailer LED lights. The relay being an electro-mechanical device that needs a current of a certain duration and strength to operate will not be triggered by the short weak pulses in the control signal, so effectively filters them out so the LED doesn't flicker, but will still turn the light on when the full operating current is sent down the wire.

So, a complete solution requires both a load resistor in parallel across the indicator circuit, and a relay in series to operate the lights - as in the attached diagram from a LandRover group.

It's worth noting that not all cars and all trailers with LED lights have these issues, some don't need this workaround, and some (like mine) only need load resistors and not relays. However, some auto sparkies will fit them both to a trailer just in case, as they don't know what car you may use the trailer with in the future, and the worst case scenario is that you'll just be using a bit more power in your indicator circuits than LED lights should, and you have one more element to fail - the relays!