Long answer: It's necessary to use some math to explain this, but I'll try to skip it.

Let's talk about holes first, because it's easier to understand. In a semiconductor, the electrons can have only some values of energy. And sometimes you have a place where an electron is "missing". In a normal crystal you must find an electron there. But perhaps there is an impurity (an atom that is not the expected atom in the crystal) or perhaps another process, so you get an empty site where an electron should be. For some calculations, it's more easier to think about the missing electron than to think about all the other electrons that are there, so it's usual to call it a "hole". For many calculations, you can use the hole as a real particle, and get the correct result. (The physicist that work with crystal truly believe that they are real particles, and they will become slightly annoyed if you argue otherwise. :) ) The holes are some kind of weird particle that only can live inside a crystal, they can´t travel in empty space. https://en.wikipedia.org/wiki/Electron_hole

If you combine an electric and a magnetic field, then you can measure the charge of the particles that are traveling inside a semiconductor or crystal. The strange thing is that if you do this experiment with a semiconductor with many holes, then your experiment will show that there are positive particles traveling there, as if the hole were real and the lack of a negative electron produce positive hole. (It's possible to get the same result using the electrons and fermionic algebra, but the calculation is very complicated, it's much easier to get alone and think about holes.) https://en.wikipedia.org/wiki/Hall_effect

Now, what is the mass of a hole? Well, it's complicated. A hole is not only the lack of one electron, it's a perturbation in the nearby region caused by the missing electron. All the nearby electrons (and atoms?) move slightly to partially compensate the missing electron. (You need even some Fourier analysis for the details here.) But the mass of the hole is not equal to the mass of the missing electron, it's complicated, it's different and it's possible to measure it in an experiment. See again: https://en.wikipedia.org/wiki/Effective_mass_(solid-state_ph...

Back to "electrons". In a crystal, the electrons are not isolated electrons, you must think more in all the perturbation in the other electrons (and atoms?) that cause an electron moving inside the crystal. Imagine that each moving electron draft/repel/whatever a little the surrounding electrons. So in the experiments you can't measure the isolated electron inside the crystal, you measure an electrons with its tar ball. Perhaps it would be better to call it with a different name, to avoid the confusion with the isolated naked electron, let's call it electron-tar-ball. You can measure the mass of this electron-tar-ball as you can measure the mass of a hole. And the mass is not the mass of an isolated electron. See again the same article.

In the article, they say that the electron is heavier, they mean that the electron-tar-ball is heavier. The problem is that physicist call them "electrons", so it's confusing.

The "electron-tar-balls" real particles, they are as real as "holes". But they can only live inside crystals not in empty space. The difference is that in some experiments you can make the additional electron in the electron-tar-ball escape from the crystal and travel in vacuum, and when it's traveling in vacuum you get the usual mass.