I came across this formula to convert from one unit of temperature to another. I could not find any proof for this though. In the formula, 'X' is the temperature value to be converted, 'UFP' is the upper fixed point and 'LFP' is the lower fixed point of the temperature scale.
The formula is just a proportionality. It says that differences in temperature are proportional on all linear scales. This is true for all linear scales, not only temperature scales.
Different scales assign different numbers to UPF and LPF. However, the same temperature measured on different scales is the same distance above LPF as UPF is above LPF. For example, the temperature at which sulphur melts is always the same proportion of the distance between the freezing point (LFP) and boiling poin (UFP) of water, on any scale.
It is similar when comparing lengths. The ratio of the height of a building compared with the height of each storey is the same whether you measure height in feet or in metres. The answer is always the number of storeys (assuming they are all the same height).
The conversion of temperatures is more complicated because temperatures are points on a scale. You can only compare distances between points (intervals) on different scales. Your formula tells you that the ratio of two intervals is the same whatever temperature scale you use.
What might be misleading about your formula is that the 'constant' is not a universal constant like the Universal Gas Constant $R$ in the Ideal Gas Law $pV=nRT$. The constant in your formula always depends on what 2 intervals you are comparing, just as the ratio of heights depends on what building you are measuring.