First, a quick refresh of the basics. Your mod can’t measure temperature. What it can do is measure resistance. TC works by using coils that change resistance as they heat up. The first step in TC is for the mod to take a resistance reading of the coil at room temperature. Then, using the TCR value, your mod can calculate the amount of resistance change for each 1°C change in temperature. With this, your mod knows how much resistance your coil gives for whatever temperature you choose to vape at.
Say, for example, your coil is 0.25 ohms at 20°C and the TCR value is 0.00138. Your mod multiplies 0.25 by 0.00138 and determines that each 1°C change in temperature value equates to roughly 0.000345 ohms of resistance change. So, if you set your mod to vape at 225°C, you mod multiples 205 (the change in °C from 20) by 0.000345 and then adds in the starting resistance of 0.25 and knows to apply power until the coil gives a reading of around 0.320725 Ohms.
The key to understanding why this tweak tool is important is in realizing that static resistance coming from outside the coil will change (sometimes drastically) the amount of change per 1°C you mod thinks is needed. If, in our previous example, we have an RTA deck that adds 0.05ohms of resistance, then your mod reads a baseline resistance of 0.30 and calculates a required change of 0.000414 Ohms per 1°C. With this, a temperature setting of 225°C works out to roughly 0.38487 Ohms. Since 0.05 of that is coming from the RTA deck itself, the coil would actually need to hit 0.33487 Ohms before the mod thinks it hit the right temperature. In reality, the coil is at roughly 266°C at that resistance level.
Now, you could just lock your mod into a lower resistance than what it reads. You can, for instance, set it at the 0.25 Ohms the coil actually is so the mod calculates the right Ohms mapping for whatever temperature you select. However, since the RTA deck is always making the reading higher than the coil really is, TC protection will kick in well before it is really needed. Likewise, you could just lower your selected temperature until the vape feels like it is at the temperature you really want it to be. This, however, is a guessing game and your mileage may vary when it comes to accuracy.
This tool simplifies everything for you by calculating an adjusted TCR value that compensates for the static resistance. The process to do this is simple. We start with the correct coil resistance and TCR values to obtain the change per 1°C just like your mod would. Then, we divide this by the coil Ohms plus the static Ohms to obtain a new TCR.
Using the same hypothetical coil as before as an example, we know that our base line coil is .25 and a TCR of 0.00138 gives us 0.000345 Ohms of resistance change per 1°C meaning a 21°C coil would have a resistance of 0.250345. If we take 0.000345 and divide it by 0.3 (the 0.25 coil plus the 0.05 resistance from the RTA deck) we get an adjusted TCR of 0.00115. By using this new TCR in your mod, it will correctly map out the 0.000345 Ohms per 1°C change needed for the coil even though it is reading a baseline of 0.3 ohms. Now, if you set a temp of 225°C the mod calculates a needed resistance of 0.370725 Ohms. Since 0.05 of that comes from the RTA deck, the coil will be giving 0.320725 Ohms, which is correct for that temperature.