It is probably a good thing to compare protein electrophoresis with DNA electrophoresis here. For DNA, you know that it has a phosphosugar backbone that comes with a negative charge at ambient $\mathrm{pH}$ per phosphate group. This means, that the charge-to-mass ratio, as the quote implies, is similar, no matter how long the DNA fragment is. (Not identical, because different bases have different masses and DNA typically has a seemingly-random base distribution.) Yet they still travel at different speeds.
This is because the gel is essentially a spider’s web for the DNA fragment. It somehow needs to find the pores of the gel and fit through. These pores have different, randomly distributed sizes and are not all aligned perfectly as you would assume for a crystal structure. Therefore, this process requies some movement perpendicular to the direction of charge. The larger a DNA fragment is, the harder it is to constantly change directions to fit through the misaligned (and potentially small) pores, so the speed of diffusion is slower. Hence, low-molecular weight DNA fragments wander far through the gel while high-molecular weight fragments remain closer to the starting point. Since size and molecular weight nicely correspond to the length in bases, markers are often labelled in terms of $\mathrm{kb}$ — kilo-base pairs.
Similar concepts apply to SDS protein electrophoresis. By denaturing the proteins with the detergent SDS (sodium dodecyl sulphate), they are linearised and equipped with a strong negative charge from the sulphate groups. Much as is the case for DNA, the number of dodecyl sulphate molecules is approximately proportional to the length of the protein in amino acids. And since they are linearised, i.e. their tertiary and quaterneray structures (and to a certain extent also their secondary structures) are destroyed, they very much behave like a DNA fragment when forced through a gel: the shorter they are the better they fit through and the faster they travel.
In the protein case, however, the markers are typically not labelled as a certain number of amino acids but as corresponding to a certain molecular weight (often given in $\mathrm{kDa}$). This is because amino acids vary greatly in size from the very small glycine to the very large tryptophane or arginine. The vicinity of one of those larger amino acids may allow for an extra dodecyl sulphate to be added when compared to the vicinity of small amino acids.
But why don’t larger proteins/DNA fragments that carry a greater negative charge move through the gel faster? Isn’t a greater force being applied onto them?
I’m glad that you asked me that. Yes, a greater force is applied. But, as you may remember from physics classes, $F=m\cdot a$, so consequently $a = \frac{F}{m}$. Therefore, the larger molecule may experience a stronger force, but due to the molecule’s higher mass, the overall acceleration is equivalent (again: not identical but similar). This allows the size-discriminating gel network to do the sorting by size.
