It's from statistical inference, e.g., when the goal is to find the values of a model's parameters that match the sample. So if the model is y = f(x, params), inference gives you params, and prediction gives you y for a value of x that you haven't seen before.
More than that, inference usually refers to acts of decision making or evidence evaluation, like testing hypotheses or interpreting confidence/credible intervals.
Also, shouldn't the verb be inferring?