I conceptually understand a result, but not its mathematical implementation – how can I defend using it? There is a particular result from another author that I wanted to use in my research which is a central component to my research question (it’s an animal’s growth rate). However, deriving the results computationally requires an understanding of some areas of mathematics that neither I nor my supervisor have had training in. The nature of my work draws heavily on objects from mathematical biology, and I’ve recognized that this is a weakness in my education portfolio I intend to fill when I do my PhD.  

I was making some progress in learning the math on my own, but I occasionally needed to reach out to the authors for clarification because I had nobody else I could speak with. In the end, the authors volunteered to give me their code so that I could produce their results and allow me to proceed with my research. 

I find myself in the conundrum that I don’t know how to reconcile or could defend when pressed by a committee: conceptually I understand what the authors are doing, but I lack the training and time to do the computations entirely on my own. Yet I have the tool I need to continue on with my research. 

I feel like this scenario is not entirely uncommon, particularly for graduate students, but I’m wondering if a reasonable defence can be formed for using another’s work (given the context above) if pressed by a committee. 
I only have an analogy that I would offer as a defence if pressed today.
How can I approach this issue?
When it comes to applying mathematical results, it is the norm that you do not understand all the nuts and bolts of what you are applying. This is nothing to be ashamed of. Instead, it’s the only way that modern science can work – it is too vast and intertwined to be understood in its entirety by one person. The crucial part is that there is a clear interface between theory and application where everything is communicated clearly.

To give an illustrative example: I recently wrote a paper where I derive some rules for building ecological models from basic principles. I did my best to describe said principles as well as the rules and their consequences so that those applying my work can use them. However, I expect that the vast majority of my readership cannot understand the mathematical proofs that go in between. I must expect that they treat this as a black box. On the other hand, in order to perform my proofs, I in turn relied on theorems whose proofs I did not understand. 
Mathematical theorems are great interfaces. They give you some statement of the form: “If X, then Y.” You only need to understand the statements X and Y, but you do not understand why one implies the other. That’s what mathematicians are for. On the other hand, the mathematicians who proved the theorems I use have no particular expertise in ecology as far as I know.

Now, there may not always be a clear interface. In that case, I suggest to create one yourself (and not only for yourself but also for your readers): Write a passage for your thesis that describes what you understand and need the tool to do. Then back up everything with references from the relevant publications and documentation. If necessary, ask the authors again.