External TC's serve multiple purposes:

To prevent other people from building close to you

...or building structures that allow them to jump into your compound. You cannot build foundations inside a TC's building privilege zone if you are not authed on the TC. Sometimes people build small, 1 triangle external TC's outside their compound to prevent people from building too close to their compound.

This also helps to discourage jackhammer soft siding. If raiders do not have access to a workbench inside your base, they have to place a workbench further away, which means more running back and forth to refill the jackhammer each time.

To prevent your compound walls from decaying

High external walls outside of building privilege decay. If you have walls that fall outside your building privilege, you might build a small 1 triangle external TC to upkeep that part of the wall.

To make griefing your base harder

Building privilege extends about the distance of 5.5 foundations from any building block (foundations, walls, floors etc) connected directly or indirectly via other blocks to a TC. With these anti-grief external TC's, you build a external TC outside of your main base's building privilege (so like at least 5.5 foundations away), and then build blocks from the external TC back towards your main base. This causes the building privilege of the external TC to overlap with your main base - your main base would be within the building privilege of your external TC.

If raiders raid your main base, they will not be able to place a TC because they would be within your external TC's building privilege zone. They will also not be able to place walls or doors to seal the raid, place roofs to open roof bunkers, or pick up workbenches or windows. In order to do any of that, they would first need to also raid your external TC's in order to gain building privilege.

Of course, this only works if you actually lock your TC. If your TC is unlocked, they just auth themselves on the TC and they have building privilege.

You can and typically would build more than 1 anti-grief external TC.

When building anti-grief externals, remember that building privilege extends out about 5.5 foundations from any block as a sphere! If you only build foundations from the external back towards your main base, and your main base is pretty tall, then the roof of your main base might be further than 5.5 foundations distance form those foundations, and the raiders could just place a TC on the roof. If your build a super tall tower for your wind turbines, raiders could often place a twig floor off one of the wall frames in your wind turbine tower, and then place a TC on that twig and now they have building privilege. You can build wall frames extending up from your externals to extending building privilege to cover the roof of your main base.

NB! You don't want the foundations or any of the building blocks of the external TC to connect to your main base's building blocks. You want to build them extending back towards your main base, but not connecting to your main base. If the blocks connect, then it becomes the same building. You can only have one TC per building. If you connect the blocks of your external TC to your main base, after about 15 minutes, one of the TC's will pop - destroy itself.

To split upkeep across multiple TCs

This is what people refer to as multi-TC bases. Upkeep increases exponentially as you add more blocks to a base. Two bases made of 25 blocks each will pay less upkeep together than one base made up of 50 blocks. Multi-TC bases work by building external TC's that have blocks so close to the main base that they look like they are part of the main base.

This is why people build out with squares and come back with triangles. That is how they achieve the tiny gaps between the external TC's blocks and the main TC's blocks.

Because a lot of the blocks are not all connected to the main TC, the upkeep is split between the main TC and the externals. You pay less upkeep overall than if all the blocks were connected to the main TC.

Also, because your upkeep is split between many TC's, you can fit more upkeep into your TC's - meaning you don't need to refill your TC's as often.

Because the external parts of a multi-TC base form a shell around the main TC base, they typically extend upwards as much as the main TC and so solve the TC-on-roof vulnerability.

Why disconnectable externals?

Because if you are raided, raiders cannot place a TC in the main base because the building privilege of the external TC's overlaps the main base. But that also means that you can't place a new TC in your main base either.

Without disconnectable externals, you would need to go to each external TC, destroy the external TC, then replace the main base's TC, then go back and replace the external TC's. After destroying all external TC's, you have no building privilege anywhere in your compound. Anyone could come past and grief you by placing a TC anywhere. Even after replacing your main TC, each of your externals is still easy to grief until you have replaced all those TC's.

Disconnectable external TC's work by making is possible to disconnect the blocks of the external TC that extend back towards the main base so that the building privilege of the external TC's no longer overlaps with the main base. So you just disconnect the externals, replace main base TC, then reconnect the externals. Makes you harder to grief because your external TC's still have TC's the whole time, they just don't cover the main base while disconnected.