The amount of knowledge one has is measured in the number of truths he can handle efficiently in some beneficial way. Truths are deduced essentially using logic even though the deduction process is not always deterministic. The depth of knowledge one has can be measured by quantifying the resources needed for the deduction process: the most important resource being time.
A person is said to have deeper knowledge than another if the other person needs more time for deduction than the former. Having deeper knowledge hence can buy you time! To increase the depth of your knowledge you need to improve your deducing abilities and for that you need to constantly practice it so that you become fast. One useful improvement technique is memoization where you do not recompute the same sub-problem every time: if you solved it once just remember and use it. Don't resolve the same puzzle over and over again. The goal is to constantly build a global lookup table for your life. This will help not only you but also those with whom you can share the table.
This is basically why experience can help increase your depth of knowledge.