A real life example of Occam's Razor in practice goes as follows:
Crop circles began to be reported in the 1970s. Two interpretations were made of the circles of matted grass. One was that flying saucers made the imprints. The other was that someone (human) had used some sort of instruments to push down the grass. Occam's Razor would say that given the lack of evidence for flying saucers and the complexity involved in getting UFOs from distant galaxies to arrive on earth (unseen and traveling faster than the speed of light I suppose) the second interpretation is simplest. The second explanation could be wrong, but until further facts present themself it remains the preferable theory. As it turns out, Occam's Razor was right as two people admitted to making the original crop figures in the 1990s (and the rest have apparently been created by copy-cats). Despite this fact, some people still ignore Occam's Razor and instead continue to believe that crop circles are being created by flying saucers.
The simplest model is more likely to be correct--especially when we are working with unusual phenomenon.
For more on Occam's Razor see Why People Believe Weird Things, The Demon-Haunted World: Science As a Candle in the Dark, an invisible, flying, fire-breathing dragon is in my garage, and the comments and links here.
For a discussion of David Hume's quote
...no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous than the fact which it endeavours to establish.see Chapter 6 in Richard Dawkins's Unweaving the Rainbow: Science, Delusion and the Appetite for Wonder.