How can Buddhist believe in Re-Incarnation, when they do not believe in soul?

Buddhists do not believe in reincarnation, the view is rebirth.

Try this. According to the TV show Madmen, there is a Japanese saying that says that a man is whatever room he is standing in. So in a business office, he IS a businessman, at home, he is a husband and a father, at a bar, he is a drunk, etc.

Now a man can walk from room to room, showing a continuity from "life" to "life". Clearly there is some sort of continuity, but when the man was in the business office, he was not the drunk, and vise versa, so we have no true business man, no true drunk. Just the appearance of them.

Rebirth says the man really is whatever room that he is standing in, and as that man he is as real as that room, and this is possible precisely because there is no true, central core man behind the causes and conditions that give rise to the specific appearance of the business man. If there were a non-empty self, it would not be subject to causality, and would not, therefore be capable of taking on an appearance, or having interactions. The appearance of a self is all there ever is, and it is precisely the clinging and grasping onto the basic notion of a true self at all that perpetuates rebirth from room to room.

To understand rebirth a bit more clearly, we can realize that at no point did we ever stop being a child, and start being an adult, but looking back, it is clear that one is different from the other. In terms of lives, we cannot say when one definitively ends and another begins, we can merely say that we can point to the apparent beginning and end based off of certain inferential and ultimately arbitrary definitions. There is never a clear moment when one room ends and another begins, never a clear moment when the businessman starts, and the businessman stops. We just "draw a line".

To go a bit further, because we cannot assert a true beginning, there is no way to assert a true lifetime to reincarnate into, there is merely the codependent origination of apparent phenomena. The whole thing is nothing but the perpetuated arising of empty phenomena due to causes and conditions giving rise to other causes and conditions, but nothing ever truly breaks over into actual true existence. We seem to perceive this, but that is just a relativistic assumption.

TL;DR - Buddhists do not believe in reincarnation or true separate lifetimes, so there is no problem, there is merely the perpetuation of the causes and conditions that give rise to apparent, yet ultimately empty phenomena.

Part 1:

The answer comes from Milindapanha 3.5.5. This means that rebirth without transmigration of the soul from one place to another, takes place like a transfer of "information".

The king asked: "Venerable Nagasena, is it so that one does not transmigrate and one is reborn?" "Yes, your majesty, one does not transmigrate and one is reborn." "How, venerable Nagasena, is it that one does not transmigrate and one is reborn? Give me an analogy." "Just as, your majesty, if someone kindled one lamp from another, is it indeed so, your majesty, that the lamp would transmigrate from the other lamp?" "Certainly not, venerable sir." "Indeed just so, your majesty, one does not transmigrate and one is reborn." "Give me another analogy." "Do you remember, your majesty, when you were a boy learning some verse from a teacher?" "Yes, venerable sir." "Your majesty, did this verse transmigrate from the teacher?" "Certainly not, venerable sir." "Indeed just so, your majesty, one does not transmigrate and one is reborn."

Here's another quote by Ven. Narada Mahathera about the transfer of "information" using the analogy of radio waves picked up by a radio set:

Rebirth takes place immediately, irrespective of the place of birth, just as an electromagnetic wave, projected into space, is immediately reproduced in a receiving radio set. Rebirth of the mental flux is also instantaneous and leaves no room whatever for any intermediate state (antarabhava). Pure Buddhism does not support the belief that a spirit of the deceased person takes lodgement in some temporary state until it finds a suitable place for its "reincarnation."

I summarized in another answer (based on the Theravada tradition) that:

Every person's habitual thoughts and actions (karma) shapes his "inclination of awareness", that becomes the last state of mind at death, and this is the "information" that gets transferred without carrier, the way flame is transferred from candle to candle, or radio waves are transferred from one radio to another.

Part 2:

The question is not clear to me. With Nirvana, one ceases to appear in a new life again in the future.

There is a difference between re-incarnation and re-birth. The Dhamma-Vinaya teaches rebirth based upon paticcasamuppada [dependent arising]. The usual religious epistemology has humans having a body, mind (conscious), and spirit (soul). The Gotama Buddha was pragmatic in his logis and the Dhamma-Vinaya is empirical. Since the existence of a soul or an inherent Self can not be evinced, we are left with the pragmatic solution of dependent arising as taught by the Enlightened One. Since the body becomes decrepit and dies what we are left with is consciousness that becomes attached to the next re-birth. It is a consciousness devoid of the last personality but still encumbered with some of the past ignorance/dispositions (and occasionally some memory of past lives).

Nibbana is not a place; like the mythical heaven of religions. Nibbana is the release, the freedom from samsara; the cycle of re-births. Everyone and anyone can attain nibbana. Once the dispositions that are the result of greed, hatred and etc. are dissipated. This is actually a much deeper topic to discuss than can be done here.