David Smith links to a video which explains how neural networks do their thing:
In R, you can train a simple neural network with just a single hidden layer with the nnet package, which comes pre-installed with every R distribution. It’s a great place to start if you’re new to neural networks, but the deep learning applications call for more complex neural networks. R has several packages to check out here, including MXNet, darch, deepnet, and h2o: see this post for a comparison. The tensorflow package can also be used to implement various kinds of neural networks.
R makes it pretty easy to run one, though it then becomes important to understand regularization as a part of model tuning.