Amelia assumes that the data follow a multivariate normal distribution, so all information about the relations in the data can be summarized by just means and covariances. When data are incomplete, Amelia uses the well-known EM algorithm to find corrected estimates of the means and covariances. See Little and Rubin (2002) for more detail.
In their original form the EM estimates cannot be used to create multiple imputations, as the estimates do not reflect the fact that they have been estimated from a finite sample. In order to solve this, Amelia first takes m bootstrap samples, and applies the EM-algorithm to each of these bootstrap samples. The m estimates of means and variances will now be different. The first set of estimates is used to draw the first set of imputed values by a form of regression analysis, the second set is used to calculate the second set of imputed values, and so on.
As Amelia assumes a multivariate normal distribution, it will work best when your data are approximately normally distributed (possibly after a transformation), and when the statistics you calculate from the data in your complete-data analysis are near the center of the distribution, like means, modes or regression weights.