I think many people are misinterpreting the question.
Divination was illegal in ancient Rome. Many Romans believed in it anyway, but it shouldn't be mistaken for the augury that was an integral part of Roman society.
The key difference is that divination is about foretelling the future, whereas augury is about determining the will of the gods in the present. Two very different things.
So to answer the question as written, you have to ignore the augury that was part of Roman society - that's not the same thing at all. You need to look at the soothsayers who often operated in Rome and other parts of the Roman republic/empire despite it being illegal to do so. They were run out of town by the authorities every so often, but as soon as a crackdown was over they would return or be replaced by others.
I suggest getting the question clarified - is it about the augury that was part of the official state business or about the divination that was illegal but frequently practised by individuals in markets and suchlike?
EDIT: As for why many Romans believed in divination, well, that's pretty much universal. People have been trying to predict the future pretty much everywhere pretty much all of the time. It's not just in the past either, nor only in primitive societies. Open almost any crappy mass-market paper and you'll find astrology in it. It's obviously rubbish, but plenty of people believe it because they want to know the future. Just as plenty of Romans did.