Telomeres are the caps of repeating DNA sequences at the ends of chromosomes. A little of that length is lost with each cell division, and telomeres thus form a part of the limiting mechanism that prevents most cells in the body from simply proliferating forever. That is fairly important: when the limiting processes fail, we call the result cancer. There are mechanisms to lengthen telomeres, however, such as the activity of the enzyme telomerase for example. Stem cells must maintain themselves capable of replication so as to ensure a continual supply of new daughter cells with long telomeres to maintain tissue health, and telomerase allows that to happen. Telomeres and telomerase do various other things too, such as influence patterns of gene expression, as nothing in a cell ever acts in just one process. That would be far too easy.
Average telomere length is most commonly measured in the leukocytes, or white blood cells, from a blood sample. This can be correlated to age and health, though there are subtleties here and some ongoing discussion on the best way to construct these measures, as well as whether or not some of these measurement mothods are in fact useful at all. The shifts in average telomere length over time may be a reflection of the pace at which stem cells are active to introduce new long-telomere daughter cells into circulation, which in turn has some relation to aging because stem cell populations tend to decline in response to the rising levels of cellular damage that cause degenerative aging. This stem cell decline is most likely an evolved balance between cancer risk and tissue failure that arose to lengthen human life span in comparison to that of other primates, a consequence of our greater intelligence and culture that allowed older individuals to contribute meaningfully to the survival of their grandchildren and thus selected for greater longevity. This is all a simplified sketch of what remains a debated and more complex picture, however. The point to take away is that telomere length looks a lot more like a marker of aging than a cause.
That all said, telomere length as measured by today's young medical startup companies is not really a particularly good measure of aging or ill health, especially taken in isolation for one individual at a single point in time. It is a very blunt tool at this point, nowhere near as accurate as, say, the DNA methylation patterns that can indicate chronological age to within a few years. It is also open for debate as to what shorter average telomere lengths for white blood cells really actually indicate in any given situation, given how dynamic they are in response to transient illness - something that may have more to do with immune system state than anything else. So it is wise, I think, to pay attention to studies of this nature that point out further issues with the use of telomere length measures in any single tissue:
The relative length of telomeres measured in peripheral blood leukocytes is a commonly used system marker for biological aging and can also be used as a biomarker of cardiovascular aging. However, to what extent the telomere length in peripheral leukocytes reflects telomere length in different organ tissues is still unclear. Therefore, we have measured relative telomere length (rTL) in twelve different human tissues (peripheral blood leukocytes, liver, kidney, heart, spleen, brain, skin, triceps, tongue mucosa, intercostal skeletal muscle, subcutaneous fat, and abdominal fat) from twelve cadavers (age range of 29 week of gestation to 88 years old).
The highest rTL variability was observed in peripheral leukocytes, and the lowest variability was found in brain. We found a significant linear correlation between leukocyte rTL and both intercostal muscle and liver rTL only. High rTL variability was observed between different organs from one individual. Furthermore, we have shown that even slight DNA degradation leads to false rTL shortening. Despite the relatively low number of individuals analyzed and the large age span of the donors, we suggest that there is a very low correlation between the rTL within most tissues and the rTL in blood leukocytes. Thus, the use of leukocytes as a source of DNA for rTL analysis for the estimation of individual "tissue age" is questionable.
Since telomerase activity and cell turnover rates vary widely throughout the different tissue types of the body, it shouldn't be surprising at all to find that average telomere length in different tissues may or may not be meaningfully correlated with either age of the tissue or between different tissues of the same age. But as the authors point out, this isn't a large sample, and they are criticizing an already fairly weak correlative tool. As is often the case more data wouldn't hurt.