The way I like to summarize it when talking to non-technical types is this: The odds of any one ticket winning the lottery jackpot are astronomically small. Regardless, people win the jackpot quite regularly.
Low probability per trial × many trials = reasonable probability of occurrence overall.
Rounding small probabilities down doesn't fully explain all the ways folks get tripped up thinking about probabilities. For example, the Birthday Paradox doesn't fit that model directly, because it's counter-intuitive what constitutes a "trial". As the number of people involved grows linearly, the number of potential pairings grows quadradically, and most folks don't really take that into account.
Extending that to the lottery example: It's far, far more likely that two people bought the same numbers than it is that anyone matched the jackpot numbers. (And that's before taking into account the fact that folks that pick their own numbers rarely pick very random numbers.) But nobody's interested in that coincidence until the folks with the same number also match the jackpot number.