Like, what exactly are they seeing when they look at the mitochrondria of each person in the study?
They're seeing this:
What is this? That's a little complicated. The short answer is - it's a bunch of segments of DNA, marked with a flourescent dye and sorted by segment length from longest at the top to smallest at the bottom.
Basically, we know that most mutations increase the length of of highly repetitive sequences already found within DNA. Using a technique called PCR-RFLP, we can amplify a sample of DNA - make a bunch of copies of it - and then cut the DNA with enzymes around the repetitive sequences.
These sequences are loaded into holes at one end of an agarose gel and then exposed to an electric field for a certain period of time. Since DNA is electrically charged, the sequence fragments travel from the holes through the gel. Since the gel is viscous, smaller sequences travel faster - longer sequences get "tangled up" in the gel. The DNA is also dyed with a florescent dye so we can see where it winds up.
The result is a series of bands representing the relative length of different people's samples of mtDNA. This represents the number, and position, of mutations in a human's maternal genetic ancestry.
Like, some question is raised about the value of a mere sample, no matter how diversely collected, considering that it's still only a very small portion of the living human population.
The odds that a random sample of human beings from all around the world all have the same maternal ancestor are
extremely, extremely low, - unless all human beings have the same maternal ancestor. It's just a matter of basic statistics.