Oldie paper but still a goodie. Some other algorithms to consider... Random Forests, Gradient Boosted Machines (similar to AdaBoost), Logistic Regression, Neurel Networks.
No advantage over taking a decent statistically valid subset and running C4.5 (or a variant with floating point splits) apart from it's less work.
But then, some models are pretty much impossible to understand or interpret.
You can, however, calculate importance scores for the features used. Brennan's original paper gives a good algorithm for doing this (in short: for each tree, permuting the data along some feature for an out of bag sample and seeing how much worse it does.)
To oversimplify, each of the many, many decision trees is going to do a poor job building a classifier. Each tree only uses a subset of the data and a subset of the features. So, the results of each tree won't generalize well (and may not even do well classifying the data it was trained on). But, for each tree that misclassifies in one direction, another tree will misclassify in the other direction. The theory (which appears to be validated in practice) is that poor decisions cancel each other out but good decisions don't get canceled out.
Although, it isn't quite a simple as a "good" tree and a "bad" tree. All decision trees are bad but there should randomly be good parts in the bad trees. Each tree is going to be "mostly crap" (as Jeremy Howard puts it). But "mostly crap" means "a little good." The crap in one tree cancels out the crap in another tree. The good parts are left behind... and... like magic... you have a good classifier that is fast although somewhat of a black box.
Jeremy Howard from Kaggle is a big believer and where most of my knowledge of Decision Tree Ensemble algorithms come from: https://www.kaggle.com/wiki/RandomForests
To the extent that the errors of the individual trees are uncorrelated, the random forest keeps the same low expected error of a single tree while reducing the high error variance of a tree. This is the same reason a mutual fund is better than a single stock -- on average they have the same return, but the latter has far more variance.