Review
I think this is going to work a bit differently for everyone, and I have no idea what kind of pre-existing math knowledge you have, but part of the answer is: play with it. Let it be fun. Realize that math isn't stuffy rules passed down by wise elders, it's something that came out of people trying to figure out a way to capture the essence of some facet of the world in a way they could work with and manage.
For me, I first really started appreciating math, as opposed to being good at what schools told me math was, when I started reading books like The Art of the Infinite (Written by the Kaplans, who founded The Math Circle, a program to teach kids a creative, problem-solving-based, cooperative approach to math). Other good books to get a flavor of what I mean by "play" might be: Flatterland or anything else for a mass audience by Ian Stewart, anything else for a mass audience by the Kaplans, and if you're up for it, the main popular works by Douglas Hofstadter (Godel Escher Bach, Metamagical Themas, The Mind's I).
Heck, go back and read Alice in Wonderland, but with an eye to the fact that Lewis Caroll was a mathematician writing drug-addled fiction about the advanced math of his day.
I would even recommend, "Surely You're Joking, Mr. Feynman" for a few really good anecdotes about what it looks like when some actually tries to use their knowledge to operate in the world, as opposed to others who are treating it as some kind of separate magisterium.
And then, try to find little puzzles, things you might never have realized you never knew but that seem like they should be solvable without too much advanced math.
Random examples:
A friend of mine once told me "if you're making a decision that depends on a number, and you haven't multiplied two numbers together, you're messing up." I think this is basically right, and I've taken it to heart.
Some triggers for me:
Verbiage
When I use any of the following words, in writing or in speech, I either look up an actual number, or quickly do a fermi estimate in a spreadsheet, to check if my intutitive idea is actually right.
Question Templates
When I'm asking a question, that effectively reduces to one of the following forms:
One thing that's been critical for me is having a hotkey that opens a new spreadsheet. I want "open a spreadsheet" to be in muscle memory and take litterally less than a second.
You could start with your everyday life. You can count things like:
For example, you could make a list of your monthly income and expenses. How much money you make? How much do various things cost? (If you do it on monthly basis, as would be typical in Europe, remember to divide the things you pay once in a year by 12. And if you do it on yearly basis, as would be typical in America, remember to multiply the monthly values by 12. Either way, there will probably be some things that you pay yearly and some things that you pay monthly.)
You could make a list of places that you visit regularly, by foot or by car. How far are they from your home, according to an online map? How much time until you really get there? What is then your average speed? Look at some other places, and try to estimate how much time would it take you to get there.
How much time do you spend sleeping? How much time do you spend at work? Commute? Exercise? What other things you do, and how much time you spend doing each of them? Make a pie chart of your life.
Look at the nutrition info of the food you eat; try to figure of how much sugar do you actually consume every day on average, every month, every year. Maybe try to split the food you eat into categories such as fruit, vegetables, cheese, sweets, soda, alcohol... and calculate how much you spend on each of these. You don't need to do the calculation by hand, use the spreadsheet, the idea is to think about the results.
For greater numbers, look at economy, or astronomy. Here, human brain is naturally quite bad at remembering large numbers; for example, you hear about "five millions" of this, and "seven billions" of that, and at the end of the day you remember that it was "six point one something", but you're not sure whether the something was millions or billions. That of course defeats the purpose of the entire effort.
The "one weird trick" that works here is to choose a different unit. For example, if you think about economy, don't use euros or dollars, instead, always think about "megaeuros" or "megadollars". For example, if the government spends $600 000 on something, remember (and write it down) as M$0.6. If you do this consistently, you will find it easier to remember the difference between M$0.6 and M$600.
For astronomical bodies, it helps to notice the relative sizes. It is nice to know the diameter of Earth and Moon in miles or kilometers, but it's easier to remember that Moon is about 1/4 the size of Earth. So when you e.g. learn about the size of Pluto, immediately check "so is it bigger or smaller than Earth?".
We innately know a lot about "the physical environment", but not a lot about math (because the math abilities of your ancestors have not been under selection pressure for any evolutionary-significant length of time). Although it is true that neuroscience has found brain circuitry specialized for understanding non-negative integers, it remains the fact that much more of an educated person's knowledge about math must be acquired through deliberate practice, which is slow, than his or her knowledge about stacking boxes or replacing an alternator in a car.
In summary, there is no analog in math of your experience of having your knowledge about the physical environment unlock just because you chose to pay more attention to the details of your the physical environment.
We all have an innate drive to understand (i.e., curiosity) and also an innate drive to try to win arguments. Unlocking those 2 motivations is the closest analog I can think of to of your experience in the factory of getting a skill or innate ability to unlock, but before you can wield math towards profitable ends, you must spend many hundreds of hours reading about math and doing math. The 2 motivations I just described merely make it possible for you to put in those many hundreds of hours with less use of willpower or discipline.
I recently started working in a setting making physical products, where I discovered that I had a whole nother section of my brain purposed for physical interactions between things and when things might fall on other things or melt or fall apart or shatter or scar or any number of other things. Unfortunately, because of my inexperience, this is a part of my brain I had to be prompted to use. I had to be asked directly: "What do you think will happen as a result of that?" in order to actually use that part of my brain.
I can't help but feel that the same thing must be going on with my math ability.
I want to apply my math knowledge to the world so I can actually get some use out of it, but I have no idea how to go about doing that.
Wat do?