Julian Jaynes[1] makes a pretty compelling argument that our unconscious mind does pretty much all of the heavy lifting with regards to problem-solving. The conscious mind just steers us and lets us remember what we are doing.
If I recall correctly an example he gives is looking at a math problem. Looking at 2+2 as an adult you instinctively know it to be 4. As a child you may have had to count on your fingers or write out the problem until the abstraction of numbers was solidified for you, and now it is just a black box that your unconscious has added to a "tool belt" of sorts.
For me I would describe the process of learning/problem solving as my conscious mind navigating some solution/problem space and figuring out the general shape of it. Once I have figured out the shape of the problem, the answer either:
1. Immediately looks like the shape of another, solved problem. In which case I just use that solution.
2. Is not immediately apparent.
In the case of #2 I just play around with different strategies that mostly don't work, but it helps me build a mental model of the problem. Then I wake up in the middle of the night with a solution, or figure it out while I am showering/eating/driving/etc.
[1] https://en.wikipedia.org/wiki/The_Origin_of_Consciousness_in...