Headlight Relay Kits

How does this system affect the tail lights? Will they become brighter?
Only if there is a problem (resistance) with the power feed from the alternator to the main splice.

A relay is an electrically powered switch. Installing a headlight relay uses the original headlight circuits (hi & low) to connect the headlights more directly to the power supply. A circuit breaker or fuse needs to be in the new circuit.

The normal path of power to the headlights is from the alternator to the main splice to the headlight switch.
The first part of the path is shared with everything else that is drawing power. So using a relay system reduces the load on that leg of the circuit when the headlights are on and other stuff is running. Say you're driving at night in the rain. Instead of 25 to 29 amps through the R6 wire and its bulkhead connector, it will be 15 to 24 amps. The effect of resistance goes down with less load, and up with more current.