Here is the best straightforward explanation of rebirth I've met so far. From the book The Center of the Sunlit Sky: Madhyamaka in the Kagyü Tradition by Karl Brunnhölzl, p. 183-185.
"Result reasons are used to establish the functioning of cause and result in general. This refers not only to outer or material causes but, more important, to the inner level of causality, which is the operation of karmic causes and results. Karma means that all our physical, verbal, and mental actions or impulses are causes that have effects in the same way any other causes do. In Buddhism, this principle of causality is also employed to establish the continuity of former and later lifetimes. In any case, result reasons infer prior material or mental causes from the observation of certain material or mental conditioned phenomena in the present that are the results of these causes. Basically, Buddhism says that the functioning of cause and effect means both that something cannot come from nothing and that something cannot become nothing. Otherwise, anything could randomly happen at any time or nothing would ever happen. Moreover, without cause and effect, all intentional actions, such as farming to produce the result of a harvest, would be completely unpredictable or pointless.
Therefore, in Buddhism, it is not really a question of just believing or not believing in the law of karma or former and later lifetimes. Rather, if we generally accept the process of cause and effect, we must acknowledge that it does not make sense to arbitrarily exclude some causal phenomena—that is, certain or all of our physical, verbal, and mental actions—from this general principle. This holds true even if we do not see an immediate result of these actions and hope to have avoided their consequences. In fact, we generally do experience the effects of our impulses, emotions, and thoughts, since our physical and verbal actions are constantly driven by them. When we plan a project or do our work, we do not think at all that our mental activities have no results; we take it for granted that our thoughts and imagination will result in visible actions and products. Also, we know very well the strong and possibly devastating effects of certain mental impulses, such as falling in love or declaring war. That it might take a long time for the effect of some action to ripen cannot be a basis for claiming that this action has no effect. Otherwise, it would equally follow that the movements of the original continents on earth are not the causes for the location and shape of the present continents, since the beings at that time did not experience the effect at present, nor do we at present observe these causes.
It would be highly inconsistent to say that some things or experiences have causes while others do not. This would also imply that there are some causes that have results and others that have no results. How could we reasonably define and distinguish between such phenomena? (In addition, for those phenomena that do not have causes, all the above absurd consequences would apply.) Whenever someone discovers the cause of something that was previously considered a random event—as has happened and continues to happen in science—the entire notion of causelessness or randomness is fundamentally questioned. Moreover, how could uncaused phenomena interact with phenomena that do have causes? If they interacted in a purely random way, even phenomena within an established causal continuum would become random phenomena. And if they interacted in a way that is determined by causes, random phenomena would enter the realm of causality. If there were, however, two entirely separate realms of phenomena, they could not interact at all.
As for the classical proof for the existence of past and future lives, we must first realize that if we accept the principle of causality as functioning in an all-encompassing way, then there have to be infinite chains of specific causes and results. For example, a tree that we see now has a beginningless “case history” of causes and conditions, each of which again entails its own causes and conditions. Likewise, according to Buddhism, the present moment of our mind does not come out of nowhere but arises from the immediately preceding moment of this mind. In other words, mind does not depend on anything other than mind as its specific substantial cause.455 By extending this backward and ahead in time, we naturally arrive at a mental continuum without beginning or end, which manifests as what is called the different lifetimes of cyclic existence. To arbitrarily postulate any starting point or a total extinction of this continuum—such as the beginning or the end of this life—amounts to nothing more than saying that something can come from nothing or something can become nothing. Yet this openly contradicts the notion of cause and result as such in the first place.
Further indications that are adduced for the existence of other lifetimes include facts such as newborn mammals immediately knowing without learning where and how to drink milk from their mothers.456 Furthermore, what would account for the immense range of differences just among human beings even at birth, such as being born healthy or with a severe disease, being intelligent or dumb, being born rich or in a slum, in a loving family or a violent one? How else could one explain that some people “have success” or get rich almost without any effort and others always “have bad luck” or stay poor even if they work hard? Why is it that some children can play complex pieces of classical music at an early age without training or excel at sports, while others are never able to do nearly as well even with a lot of training? Even conventionally, none of these facts can be sufficiently explained by causes that can be found in this present life, but this usually just leads to subsuming them under rubrics such as “fortune,” “fate,” or “talent.” The most fashionable category these days seems to be that “it’s all in the genes.” This is not the place to discuss this issue in detail, but if we just consider how little the genetic code of human beings differs from that of chimpanzees and some primitive worms—by just 1 percent and about 30 percent respectively— it is quite amazing to assume that the genes alone can serve as an explanation for all the differences between humans and other beings. To be sure, these differences do not consist of only physical features, but include the entire range of the human mind and its expressions, such as culture, science, philosophy, and religion, not to mention all the mental and behavioral diversity of human beings themselves, who have even less genetic variance from one another."
"If the Buddha-Nature is seen, there will be no seeing of a nature in any thing."
(Nirvana Sutra, T12n374p521b3; tr Yamamoto)
"Neither cultivation nor seated meditation — this is the pure Chan of Tathagata."
(Mazu Daoyi, X1321p3b23; tr. Jinhua Jia)
“Don’t rashly seek the true Buddha;
True Buddha can’t be found.
Does marvelous nature and spirit
Need tempering or refinement?
Mind is this mind carefree;
This face, the face at birth."
(Nanyue Mingzan: Enjoying the Way, tr. Jeff Shore; T2076p461b24-26)