A number of numerical methods have a
Method option and reading the documentation about it could give you some clues. But there are many other options depending on particulars of the functions you are interested in. What can you do with those clues? Here my answer.
SmoothKernelDistribution for example. The bandwith selection parameter has several options. One of those is "SheatherJones". If you search, particularly in google scholar, using terms like like "kernel bandwidth Sheather Jones" (here) your first hit (most likely) is "A reliable data-based bandwidth selection method for kernel density estimation - SJ Sheather, MC Jones" which describes that method. And with a little bit of luck you may find a survey that explains most of them!
However, Mathematica is a closed-software system and, of course, you cannot be sure that what you find is implemented verbatim.
Some other built-ins are actually quite vague, like
Integrate. It barely says that most indefinite integrals in standard tables are implemented. How is that implemented? This is core Wolfram's intellectual property.
NIntegrate has lot more info, which with the recipe above will point you to quite a number of interesting research in numerical analysis, if you like the topic.
Bottom line, if you can, use google scholar, even bookmark it - but other than that, Mathematica is closed software, as clearly explained by Wolfram Inc.