We only have to show that there is a statistical connection between the two events, and that there is no other plausible natural explanation. We don't have to explain how or why.
There is a natural explanation, but the philosophy of science precludes science investigating it. Let me explain with an example, which I have done on many occasions. If I had a dream, which we all have had, there is no way, using the philosophy of science, to scientifically prove I actually saw the details, I describe in my dream.
This is because we don't have the needed tools that allows others to see my dream in the second or third person. The philosophy of science requires a number of independent investigators need to see the same thing, with each able to reproduce the results in the lab, independently. Dreams do not fit in that definition since each session may be unique plus there are no tools to see directly.
As such even though billions of these dreams data point are generated by human brains each night, this subject is called soft science. There may have been more dreams generated than fossils for evolution. The latter is accepted since this can be done in the third person.
The reason for this paradox, is the philosophy of science was designed to factor out human subjectivity. If we were all looking at the forest at night and we wanted to define it objectively, but some see spirits and others see big foot, one way to approach this, would be to factor all things only seen by one person, until what is left is what we all can agree we all can see; trees and birds. This was important to science, but the subjective aspects, that were factored out also contain naturally generated data like dreams, even though dreams have billions of data points.
I agree with the scientific method when applied to the world around us, but there is also an internal world within the brain/mind that can be observed by the conscious mind, such as dreams, visions, intuitions, feelings, hunches, which do not fall under the method. Reincarnation was based on this internal data collection, which could be vast for an ancient genius devoted to a lifetime of meditation. It would still be called soft science based on the method science. But that should not be confused with poor observation, since science is not designed to deal with this data.
Part of the problem this creates is since we can all dream dreams, but since this is soft science at the best, how does science calibrate the mind to make sure they are not projecting personal soft science into science and calling it hard science? Reincarnation works this way, in the sense that real generated internal data, outside the philosophy of science, is superimposed with the reality around us. The disconnect is connected to the philosophy of science not able to comment on the internal data, since the philosophy does not allow for it.
The workings of the human mind will be the last frontier, since its exploration would require a major modification of the philosophy of science. It would need to take into account unique data by one reliable witness, with trends appearing among many witnesses. But that can only occur after all the tangible things are defined by the current philosophy, then these can be factored out to isolate the internal data. But for those of us who are curious about the inner workings of the mind, we really don't need permission to investigate, but will nevertheless have to accept the reality that the philosophy of science will make it taboo like they are a primitive religion denying data. Religions help to maintain these inner data until the day they will be investigated.