I believe in intuitions and inspirations...I sometimes feel that I am right. I do not know that I am. - Albert Einstein
Rather than just being an uncategorized and random repository of raw data, our brains excel at forming tentative relationships between bits of information. As increased experience triggers similar associations, the connections become stronger. As you can imagine where this is headed, what we're discussing is the basis of recognition.
What makes the process sometimes seem almost mystical is that we are incredibly complex machines gathering, processing, generating, interpreting, storing, recalling, merging, associating, forming hypotheses about data and relationships, and adjusting these on a pretty much constant basis. The data can be blatant and obvious, such as reading instructions for a procedure with a very specific predicted outcome. It can also be incredibly subtle, such as the warmth or feeling of humidity in someone's handshake, dilation of their pupils or the direction their gaze follows when making an assertion. It can be the difference in the time it takes somebody to answer a yes/no question, as in perhaps a child hesitating an extra half second before answering no when asked, "did you take the last cookie?"
Odors or sounds that may be too subtle for conscious recognition are quite possibly stored away in memory with a weak association to an event just experienced; if this happens twice, the association becomes stronger - but may still not surface as an explicit memory. Presented with the odor or sound again, even at a level barely above subliminal, it's quite possible to have the expectation of the event triggered - at which point we have something that looks and feels like an intuition that something is about to happen.
This is an extremely simple example, and in "real life", there are thousands of subtle data inputs that get filed away in our brain, most of which we don't consciously recognize or recall. However some degree of association and categorization is still made. Repeated or similar experiences, however subtle, may reinforce neural pathways, and the basis for intuitive flashes of insight increases.
Those of us with "strong intuitive" skills most likely have more effective ways to receive, categorize, associate and store raw data. They may be faster at associating new inputs with stored associations. Additionally - an important consideration is the ability to "abstract" meaning from raw input data. In this context, that is the ability to extract the underlying meaningful characteristics of an event or association, absent the specific details of a single specific event. Some of us are much better at abstraction than others, and those who can abstract relationships most effectively will also be those who will recognize, or possibly intuit, something about a new situation faster than others will.
Bringing this back around on topic - there's no unassailable evidence to date that anything mystical is going on when we talk about intuition, and intuitions can certainly be wrong. However we can start to appreciate a valid scientific basis for intuition, and start to allow for its potential value in decision making.