...I do not know where this comes from...
It looks like that's not a scientific article, so it doesn't have references at the end. Read it carefully, and see if it references military or industry standards for vibration dosage. Then read those standards and see if they refer back to any studies.
It'll be "fun", like black-water plumbing. Think of it as the intellectual equivalent of cleaning a blocked sewer line.
I do not understand what the plot means and how to get weightings from it. But when I think about it, I feel like it has something to do with the acceleration data collected over 0.5 - 80 Hz and get the weightings from there. The thing is the weightings are not constant in the plot for the mentioned frequency region.
Probably you get the frequency spectrum of your accelerations in $x$, $y$ and $z$, then you multiply each of those by the individual weightings shown, then you calculate your $VDV$ from that.
I was going to say you could use Parseval's Theorem and just do the computation in the frequency domain -- but there's that $a^4$ term in there, so you may just need to weight it in the frequency domain, convert to time domain, and do the math.
But this smells strongly of being an established field, which means it has established methodologies; you should not invent your own method. The established methodologies are often different from what an expert in signal processing would do, in detail if not in principle, and sometimes that different can be significant. Even if your own special way is better in every regard, it'll still be "wrong" because it'll be different from how everyone else does it.
So find the source material via a literature search: either dig into the scientific literature (starting with the references in your second paper), or if you're trying to meet some regulation into the relevant industry or government standards. Somewhere in all the muck the method will be spelled out. If it's a regulation, it'll be written in techno-legal, and chances are you'll need to find an explanation of it someplace. If you can't understand the actual defining document, then you can search on the title or designation of that document, i.e. "how do I implement SAE-234566-3?".
Also, how do I use the weightings with the raw-data collected to get frequency-weighted acceleration data?
Explained above -- convert your measurements into a spectrum, multiply said spectrum by the weightings in your graph.
Do I multiply the discrete raw-data with the the corresponding weights for each axis to get frequency weighted acceleration?
Yes. Then, presumably, you take the weighted acceleration vector at each frequency point and you get it's Euclidian length, and it's that Euclidian length that you sum up.