It's actually a collection of theories known as the mitochondrial theory of aging: https://en.wikipedia.org/wiki/Mitochondrial_theory_of_ageing

It's a little confusing to look into, because there's a bunch of separate theories about the nature of aging that all involve mitochondria in some way. You will find news articles saying that the mitochondrial theory of aging was discredited because of some study done, but when you look into it, it turns out that what was shown was that some specific variant of the theory was insufficient for accounting for all forms of aging, which is not the same thing. Each mitochondrial theory of aging is a theory about one pathway by which mitochondrial function or dysfunction results in aging damage, the reality is that many or all of these theories are true and aging is the aggregation of damage from all of them, and more pathways we have yet to discover.

Generally speaking, the vast majority of aging damage comes, directly or indirectly, from the accumulating damage from healthy operation of mitochondria over long periods of time, or the accumulation of cells with unhealthy mitochondria that produce damage more rapidly. The ELI5 is that mitochondria produce free radicals, free radicals chemically alter basically anything they touch, and aging is simply the slow accumulation of intercellular and intracellular damage, and if you follow the history of these molecules back to when they diverged from being in a healthy state, it is almost always the result of oxidative damage (e.g. free radicals of the sort produced by mitochondria).

Another ELI5 way of looking at it: you may not know this, but mitochondria only live a couple of days. They are constantly being refreshed in your cells because of the severe oxidative stress they undergo. They also sometimes break or leak, letting those reactive oxygen species into the cell and causing damage. Aging is the accumulation of this damage.

But I said aging was "downstream of" mitochondrial dysfunction. That's because not all aspects of aging is due to reactive oxygen species leaking out like I seemed to claim above. That's just one example. There are cells in your body that have lost all mitochondria, often due to a freak genetic mutation in the mitochondrial DNA of that cell. Surprisingly these cells don't die, but rather switch into a mode of operation where they slow down and live off energy extracted from the intercellular medium and converted into ATP by various molecular systems embedded in the cell membrane. These processes, as it turns out, free radicals out of the cell during operation, spewing reactive species into the body. This ends up being responsible for hardening tissue, lack of energy, and many other symptoms of aging. But the root cause? The mitochondria stopped working in that cell, so still a mitochondrial issue.

Or, the aging of heart cells and the hardening of arteries is largely due to the collection of dysfunctional lysosomes that are full of garbage they are unable to break down. These clutter cells, harm their efficiency, and eventually have enough collective effect as to make the tissue as a whole less viable. Leading to heart attacks and other cardiovascular disease, which is the leading cause of age-related death alongside cancer. Want to guess what these defective lysosomes are full of? Mostly undigested mitochondria, specifically the highly damaged structures of mitochondria that suffered too much oxidative damage from long operation.

Oh, and what about cancer? Well cancer needs A LOT of energy to keep replicating, and so it should be no surprise that many of the mutations among common cancers have to do with genes in the nucleus affecting mitochondrial function, or the various signaling pathways between the nucleus and the mitochondria of the cell. This article covers some of the ways that cancer uses mitochondria: https://pmc.ncbi.nlm.nih.gov/articles/PMC4371788/

The best lay introduction I know is "Ending Aging" by Aubrey de Grey and Michael Rae. The book is meant to be an enumeration of all the things that need to be done to biologically reverse aging, but ends up being more than 80% about mitochondria and mitochondrial dysfunction, because of its out-of-proportion impact on the aging process.