What's the point of the threshold in a perceptron?

Actually, you'll just set threshold when you aren't using bias. Otherwise, the threshold is 0. Remember that, a single neuron divides your input space with a hyperplane. Now imagine a neuron with 2 inputs X [x1, x2], 2 weights W [w1, w2] and threshold TH. I.e., this is your hyperplane equation that will divides the input space. Notice that, this neuron just work if you set manually the threshold.