Why do Buddhists believe in reincarnation? What makes a person believe in the idea of reincarnation? It can’t be “because the books/teachers say so”, given the Buddhist ideology of “come and see by yourself”
There is rebirth in Buddhism, for sure, but rebirth of what?

When we say "rebirth", we mean that something that we can peg onto as YOU, is born again after physical death. But what is YOU? The soul or self or the SAME consciousness that wanders on throughout this lifetime?

The Buddha taught that all phenomena is not self. The Buddha also taught that all conditioned and/or compounded things (including consciousness) is impermanent. It is not the SAME consciousness that wanders from childhood to old age. It changes. (See MN 38)

So, in Buddhism, there is no rebirth of the self or consciousness.

In fact, if a person injures his brain or experiences dementia or amnesia or coma, then his personality, his memories and even his consciousness could be impaired or non-existent for the remainder of his physical life. In this case, even in his lifetime, we cannot guarantee that he is totally present at every moment. Then, what more for the next lifetime?

So, if a soul or self or consciousness is not reborn, then what is reborn?

There is definitely rebirth in Buddhism, but what is reborn?


Suffering. The continuity of suffering, regardless of physical death, is rebirth.
The continuity of conditioned processes when the condition for them to exist continues, is also rebirth. For e.g. karmic cause and effect, and its relationship to an individual's perceived continued existence in this lifetime. Also, the physical and mental processes of the five aggregates, for e.g. the "stream" of consciousness (MN 38) and the thought "I am the thinker" (Snp 4.14). The continuity of conditioned processes doesn't necessarily end at the death of the physical body.


The Theravada perspective is that at every moment there is death and rebirth. In between, we have the continuity of conditioned processes, that bridges every moment. With the continuity of conditioned processes, we have the continuity of suffering, so long as ignorance is present.

When ignorance is permanently ended, craving is permanently ended and suffering is permanently ended. This permanent ending of ignorance is referred to as enlightenment in Buddhism.

In this answer, Yuttadhammo Bhikkhu wrote:


  ... the conclusion I make is that it's not that Buddhists believe in
  rebirth, it's that we don't believe in death - the latter being merely
  a concept referring to the change from one set of experiences to
  another. True death only occurs either at every moment or at the
  experience of nibbana.


Usually when people talk about rebirth, they imply that the SAME consciousness wanders on in this lifetime and onto the next lifetime. In the following quote from this answer, we read a nice explanation by Damith, of why this is not true.


  Stream of consciousness OR Continuity of consciousness means
  individual consciousness which arises (uppada), exists (titi), ceases
  (bhanga) rapidly as an effect of many causes. It's just a stream, not
  a long lasting single one.
  
  Think of a stream of water. When you look at a single point in the
  stream, you see many water droplets appear, exist, and disappear. Take
  a single droplet out of it which passed by(*) that point. If you
  consider the past, present, and future of that droplet, you'll see a
  stream. In reality, it's not the same droplet nor another droplet you
  see in the past, present, and future. Stream of consciousness is
  analogous to this example. Please try to understand it in that way.
  
  (*) Note: "passed by" - This phrasal verb can be only used when we
  talk in conventional truth. The reader may think that rūpa
  (form/matter) can move in space which is not a true in ultimate
  reality. Neither any nāma (name) nor rūpa can move. What happens is
  the arose rūpa existed and ceased then and there. After that new rūpa
  arise nearby. This is called "Deshāntarotpatti".


I'm not saying that there is no YOU (self) at all. There is a YOU, but it's not a permanent YOU. It's changing from moment to moment, depending on conditions, and compounded by different mental and physical processes.