Coulomb's law in vacuum is generally stated as
$$F = \frac{1}{4\pi\varepsilon_0} \frac{Q_1 Q_2}{r^2}.$$ However, apparently, by going to Gaussian units, $\varepsilon_0$ will take on a value $\frac{1}{4\pi}$, thus the law will be reduced to
$$F = \frac{Q_1 Q_2}{r^2}.$$ I can't wrap my head around this! On the two sides of the equation, there are quantities of different dimensions! Even if the numerical values are equal, it feels like we are throwing dimensional analysis away.
In SI units, the unit of charge (the Couloumb) is dimensionally independent of the units of length, mass, and time. In a fundamental way, charge is thought of as "its own thing", independent of these definitions. This appears to be the way that you're thinking about the familiar SI equation in the first line.
In contrast, in Gaussian units the unit of charge is effectively a "derived unit" from the units of length, mass, and time. There is not a separate definition of current or charge. Indeed, from the above equation we can see that the dimensions of charge squared in Gaussian units are $$ \text{[charge]}^2 = \text{[force][distance]}^2 = M L^3 / T^2 $$ which implies that the dimensions of the statcoulomb in Gaussian units can be derived from the units of mass (g), length (cm), and time (s): $$ 1 \text{ statcoulomb} = 1 \text{ g}^{1/2} \cdot \text{cm}^{3/2} /\text{s}. $$
The only change is in the constants of proportionality.
