The Buddha remembered clearly many of His past lives. Even today, many Buddhist monks, nuns and others also remember their past lives. Such a strong memory is a result of deep meditation. For those who remember their past life, Rebirth is an established fact which puts this life in a meaningful perspective.
The Law of Kamma can only be understood in the framework of many lifetimes, because it sometimes takes this long for Kamma to bear its fruit. Thus Kamma and Rebirth offer a plausible explanation to the obvious inequalities of birth; why some are born into great wealth whereas others are born into pathetic poverty; why some children enter this world healthy and full-limbed whereas others enter deformed and diseased... The fruits of bad Kamma are not regarded as a punishment for evil deeds but as lessons from which to learn, for example, how much better to learn about the need for generosity than to be reborn among the poor!
Rebirth takes place not only within this human realm. The Buddha pointed out that the realm of human beings is but one among many. There are many separate heavenly realms and grim lower realms, too, realms of the animals and realms of the ghosts. Not only can human beings go to any of these realms in the next life, but we can come from any of these realms into our present life. This explains a common objection against Rebirth that argues "How can there be Rebirth when there are 10 times as many people alive today than there were 50 years ago?" The answer is that people alive today have come from many different realms.
Understanding that we can come and go between these different realms, gives us more respect and compassion for the beings in these realms. It is unlikely, for example, that one would exploit animals when one has seen the link of Rebirth that connects them with us.