I have an undergrad degree in computational mathematics (though that was about 10 years ago), and spent my professional career in software development.
If I wanted to understand what's happening behind the scenes in ML, and not just blindly apply equations, what disciplines would I need to study? Stochastic calculus? Something else?
A list of courses and recommended reading would be the most useful answer, with the goal being to learn over the course of a few years, not a quick fix.
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Nowadays in a lot of good CS programs there are classes directly on statistical/machine learning. The ones that have stats probably have a bit more of the underpinnings you are looking for. Also there are theoretical machine learning courses where they prove things like VC-dimension for classifiers and what it guarantees in terms of classification performance. Bishop has a good book "Pattern maching and machine learning" that's used at Carnegie Mellon University (one of the best ML programs) that explains a lot of ML concepts instead of just giving equations, and ties in Bayesian stats views.
Books: Hastie et al., Elements of Statistical Learning is a grad-level text, 745 pages, pdfs online.
See also stats.stackexchange.com/questions/tagged/machine-learning references .
Learning by doing, in Python: scikit-learn has many algorithms, examples, solid code -- highly recommended.