Are probabilities a fact or an assumption? I noticed that a lot of textbooks define probabilities of an event 'A' to be between 0 and 1 (both inclusive).
Do probabilities necessarily have to be between 0 and 1 or is this an assumption we make to make everything nice ?

By definition, a probability is between 0 and 1, 0 meaning impossible and 1 meaning certain. (Well, for infinite sets of events, this is not quite accurate, but let's not go there.)
Your title raises other issues. In quantum theory, the probablity of an event is a fact about the world. In Bayesian methods, the probability is a measurement of the investigator's doubt or certainty. The same body of theory can be applied in either context.
