How can temperature be calculated given relative humidity and dew point?
To understand that formula, it's better to start from the more intuitive dependence of dew point with temperature and relative humidity, as illustrated by the following graph from Wikipedia:



As for any solvent with a solute, the higher the temperature of the solvent the more solute it can hold. That's why hot water can dissolve more sugar than cold water. Or, for a given quantity of sugar, hot water dissolves it faster than cold water.

The figure above says basically the same: Hotter air dissolves more water. Condensation happens when the amount of water in the air is more that the amount the air can actually hold. The dew point is the temperature at which condensation appears. Therefore, for a given relative humidity, the hotter the air, the higher is the dew point.

As you can see, in the Magnus approximation, the relationship is just a straight line, with a slope and constant changing for different values of the relative humidity.

I'll assign letters to each constant for simplicity, so that

$b=17.625$

and

$c=243.04$

With that, the dew point can be calculated as

$TD= \large c \LARGE \frac{\ln\left(\frac{RH}{100}\right)+\frac{b T}{c+T}}{b-\ln\frac{RH}{100}-\frac{b T}{c+T}}$

(see Calculating the dew point)

I won't do the algebra, but if you rearrange that formula so that $T$ is written as function of $TD$ and $RH$, you will go back to the formula you presented.