I have a 2 tier-ed personal response to this question.

For "basic math" up to high school algebra and such, having an intuitive understanding of this lets me quickly do back of the envelope math anytime I hear something that sounds off, or something matches a "pattern" of a type of math I've done before.

Two concrete examples of this are:

  - doing drop rate calculations in games (probability)..i.e. I've killed 500 of these, and it is interesting to note that I should've gotten this drop 93% of the time. I'm mathematically unlucky. 

  - trying to calculate how much it is worth to me to drive at 80mph vs 55mph to get to work, and the relationship between gas prices and my willingness to drive fast (algebra, physics, etc)

For more advanced math, I personally find my high level understanding of the topics enough to appreciate how much goes into everyday things. The algorithms that govern our understanding of radio signals, information theory to pack bits into a given bandwidth and frequency, algorithms and math to ensure what was transitted is what is received, and how hundreds people can be using it at once and still get their data where it needs to go. The amount of math that goes into this all makes me appreciate the beauty of it all.

I have no real answer to this.

People who I know who are actually really good at math (and are paid for it) say that there is no shortcut to interest and wrestling/thinking about the problems. You have to mule over things. You have to pack and unpack it over and over in your head.

If I had to put it in a phrase, it would be, "there is no shortcut to understanding without thinking".