A tail light would just connect to the appropriate power source on your bike, directly, so that it turns on when you turn the bike on.
There are many kinds, and voltages they need, and amounts of current they'll need to operate. Anything bright enough to be useful will take more power than you can get out of any of the controller's connectors; it would probably damage the controller (or the display) to connect them to it directly.
If you use 12v (automotive, motorcycle) types, you'll need a DC-DC converter that takes whatever your main battery voltage is and converts it to 13.6v-15vdc, whcih is the range that these types actually expect (12v isn't enough to run them properly). It will need to be capable of enough amps (A) to run the tail light and anything else you later decide to add (turn signals, other headlights, accent lights, downlights, etc).
If you use the types that have a wide range voltage input, intended to run directly off an ebike battery, then you don't need the DC-DC as long as your battery is within it's range.
In either case, you could connect a separate switch that connects the DC-DC's input or the wide-range light's input to the battery output. If the battery has it's own switch, and you want hte light on all the time the battery is on, you can skip the extra switch.
If you want the controller's light button to turn the taillight on too, you'll need to add a relay driver to the headlight output, that then controls a relay that turns your taillight on and off (connecting directly to the headlight output will probably blow up the internal control electronics of the display/controller and the headlight wont' work either). If the relay driver takes enough current to run, then you'll also need to use the relay (or a separate one) to control the headlight, too, and not run it directly off that controller output. THey don't usually handle much power very well.
There are other voltage lights like those that run on AA or AAA batteries, etc., and you'll need to use a DC-DC (or separate battery) to power those, like with teh 12v lights.
If the tail light you want to use also has a brake light, then you can use a relay driven by the ebrake lever to control it. The actual controller ebrake signal line probably can't handle the current needed by a decent brake light, so you don't want to directly connect the light to the ebrake wires.
The "best" way to splice the brake light control signal into the ebrake wiring is to use an extension cable that has male on one end and female on the other, to go between the ebrake connectors (whichever they are) and the controller. Then you can cut into the extension cable's housing and first test teh wiring to find which is the one you need to connect to, and then splice into that from your relay driver. That way you aren't messing with the original wiring's waterproofness, and can easily bypass a problem with any addon wiring by just taking it out of the circuit path and plugging the original connectors back together.
Some brake lights use a grounding signal to turn them on, and some require a voltage to turn them on. You'll need to know which one is required before draw up and building (or buying) the relay circuit for it.
There is a thread by Teklektik about his 2WD cargo bike (Yuba Mundo, IIRC) that has details on his lighting and wiring and parts lists, etc., that you may find a useful reference.
https://endless-sphere.com/forums/viewtopic.php?t=36959
There are also posts helping various people with this kind of thing that you can find by searching my posts, that may have useful info.
Note that if you don't know what the blue connectors are for, I don't recommend doing anything with them until you find out. Tracing where they go to and come from in the bike wiring harness is your first easy check, just folloiwng the cable from each connector to whatever is on the other end. What the actual signals are may be guessable from there, or you may then need to open up the cable or the device at one end or the other to measure things. (measuring anything without it all connected doesnt' always tell you what is really happening).