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Can anyone explain how the "floats" that are the output of the last layer correspond to the individual digits?

The largest float in the output layer (while the graph is yellow) is the activation. The largest activation in the final layer is the network's "guess". The guess is the index of the last layer, which corresponds to a particular digit.

Awesome, I just flipped that switch at the top and can see how it calculates the individual handwritten inputs. Great demo!

So I guess during training you're telling it that correct answers should be 1 and the incorrect answers should be 0.

Do the encoding choices that you make regarding the input / output of a neural network influence its performance at all? Maybe for MNIST the way you have it is the most common approach?

Usually the number of nodes in the input and output layers don't affect things all that much. They are relatively set based on the problem. The number of nodes in the hidden layers, the number of hidden layers, and various other parameters such as cost and activation functions, are mostly what you use to tune performance.

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