Letter Combinations of a Phone Number
There are plenty of solutions online and in books. However, I’m
  curious to know about the fundamentals of this problem.


Basically, this problem is trying to reveal your ability to solve problems using recursive algorithms/divide&amp;conquer for generating all possible permutations (brute-force). There are some filtering techniques that can be used in practice, but the worst-case running time is exponential.


  Is this problem essential asking for the Cartesian product of multiple
  sets? If so, what are some efficient algorithms to compute this?


Yes, one way to look at the problem is 'Cartesian product of multiple sets'. Given a string S, you can consider two sets a[i] and b[i], where a[i] is the set of character strings that can be made by substring S[1:i], and b[i] is the set of character strings that can be made by substring S[i+1,end]; then output the Cartesian product of a[i] and b[i] for every possible i. 
However, this algorithm is not the best one.


  Some solutions reference DFS and BFS. How are DFS and BFS related to 
  this problem (are they possibly referring to the type of recursion being
  used to solve the problem) ?


These algorithms use DFS for its recursive nature. Probably, it uses BFS to avoid stackoverflow.