Statistics often involve expressions that are really hard to calculate analytically. For instance, distribution functions.

Monte Carlo lets you calculate a distribution function empircally.

Rather than covering a balckboard working out math, you get a computer to sample the distribution using random selection, then base the distribution on the histogram of the sample.

For instance: you want to know the mean and standard deviation of rolling five twenty-sided dice.

You can calculate this by hand, but the exact expressiin involves lots of summation symbols and can be a pain to work with.

You can also roll five twenty-sides dice ten thousand times, make a histogram of the results, and find the mean and standard deviation of the results.

The latter is the monte-carlo way.

This is often used for parameter estimation. Certain algorithms can ensure the sampled distribution converges to the desired distribution. Thus you can get an approximation to a function that would otherwise be inpossible to calculate.