For me, the intuitive part is how mitochondria can be created by the body if it's an external structure that's been found. If I take a coin and put it in my pocket, my children don't have any coins at birth. They have to find one themselves. Why is it different for mitochondria?
An interesting thing about mitochondria too— we receive all of our mitochondria from our mother. So for every human, there is an unbroken line of direct mitochondrial ancestors leading all the way back to the first biologically unique Homo Sapian women
But to continue the analogy, it's more like you split yourself into two smaller "you"s, and in the process each new "you" gets half a coin in the pocket of each new "you".