*But I understand all these things and I don't know a damn thing about math. How do you explain that? I'm not asking to be snotty.*

How well do you understand them? Could you derive the results on your own, or double check the results of someone else? If so, you obviously know more math than you say you do. If not, then how do you know that they are right? Trusting experts can be dangerous, and it basically comes down to who you view as experts. When the general public isn't able to verify things, news reporters and the government and unfortunately even scientists resort to inaccurate generalizations or metaphors that confuse or mislead people. Heck, when the general public doesn't know enough about mathematics and science Fox can air a show about the "fake moon landing" and get away with it.
*Also, a lot of your examples may have light shed upon them with the judicious application of math, but you fail to factor in human emotion and indoctrination. It doesn't take much of a grasp of math to see that American deaths due to terrorism are a tiny percentage of overall deaths, and the number is greatly outshadowed by the number of American deaths due to, say, car accidents or heart disease or even Americans shooting other Americans.*

It took Ralph Nader to get seat belts and other safety features in cars, not just raw statistics. The reason is that the car companies did have the statistics, and people didn't. They concluded that cheap cars without safety features cost them less to build and made more money, even including the threat of lawsuits. People still haven't figured out why terrorism in America is not a greater threat than the erosion of liberties, mostly because they can't equate the number of people who could possibly have their lives ruined by witch hunts with the number that might be killed due to terrorism. It's difficult to argue persuasively using statistics and probability if the audience thinks 72.8% of statistics are made up on the spot and couldn't check them if they cared. It's just another appeal to authority in the mind of the general public.

*Just speculating, but I think a large part of my ability to grasp these things, while not really understanding the math, comes from my father's devotion to science (he holds a Master's in physics), and his drumming into me the usefulness of logical and critical thinking and the scientific method.*

The scientific method requires quantifiable results to be testable, and quantification requires mathematics. Knowing how to properly apply statistics to science is a field in itself, and understanding it is vital for valid scientific work. My guess is that you do understand more about math than you think, you just think about it in terms of logic rather than mathematics. In reality, they are equivalent in a rather beautiful way. Godel numbering allows any statement in logic to be translated into numbers such that the rules of logic are simply arithmetic operations, and proofs can be checked by algorithms. Logic is used to build mathematics out of set theory. It's really all the same bucket of ideas with different representations and rules. If you want to learn about these ideas, a fun and interesting book is Godel, Escher, Bach. It basically takes the reader from set theory through Godel's Incompleteness Theorem with lots of interesting stops along the way. It even offers one of the most reasonable (to me) explanations of what modern art is.

I agree with you that simply presenting the rules of mathematics without an explanation is probably not the best way to do it. The problem is that the steps from pure logic and set theory to mathematics are rather long and complex, and in reality you have to know some mathematics to know why the steps are useful or desirable. What aspects of mathematics did you find most difficult? From the rest of the thread it sounds like equations in general (the quadratic equation in particular) and division pose some of the hardest to understand.

The best example of the quadratic equation I can ever come up with is that of stopping a car to avoid a collision. The most pertinent result of solving the physical equation of motion for a braking car is that the stopping distance is quadratically related to the speed, so that means traveling twice as fast requires four times the stopping distance. It's also why speed limits have fine gradations at the lowest speeds: It takes almost twice as far to stop at 35 MPH than at 25 MPH. It only takes 1/3rd the stopping distance dropping from 25 MPH to 15 MPH, roughly 1/6th the distance at 35 MPH.

The reason long division has a remainder is that division is only a valid algebraic equation when q*d + r = p, where q is the quotient, d is the divisor, r is the remainder, and p is the dividend. Using the equation for division it's easy to see how the remainder is equivalent to the fractional part of rational division: (q*d + r) / d = p / d = q + ( r / d ), which results in the whole number quotient plus a fractional part. This is because q is calculated by the long division algorithm to be the greatest integer such that q*d is less than or equal to p. This means p - q*d is also less than d, which means r is less than d and r / d will always be greater or equal to 0 and less than 1, e.g. the fractional part of the division. Like lots of people said, figuring out that the long division algorithm actually produces a q with the desired property is the hard part to prove or recognize as a child. In fact, it's not true in general because q*0 + r = p has no greatest value of q that satisfies the equation q*d < p. All q satisfy the equation, so the result of division by zero is taken to be undefined.

Hopefully you can see the logic that's used in deriving the above. Mostly, when construction the set of natural numbers from set theory, the axioms and theorems are concerned with proving two things: The set of natural numbers exists and is infinite (every natural number n has a successor,n+1 and no number's successor is zero which gives a basis point for induction proofs but disallows negative natural numbers), and that every operation of arithmetic is a well defined function from a pair of distinct natural numbers to a unique natural number. For every two natural numbers addition, subtraction, multiplication, and division must be well defined so that the answer is one natural number, with the exception for subtraction and division that the result must exist in the set of natural numbers. So subtraction is only valid for a - b if b is less than or equal to a, and for division there must exist natural numbers such that q*d + r = p, and only one q and r are the correct answer for any choice of d and p except d=0. Basically, the rules at this point are so simple that they're pretty much common sense. The power comes from being able to prove every result beyond arithmetic using only those initial rules (probably a couple I didn't mention) and the axioms of set theory. The integers are a quick extension to the naturals using negative numbers, and the rationals and reals and complex numbers follow by keeping in line with the rules for distinct inputs and unique outputs for operators.