The Rule of Three is just doing the same thing you are doing: if $\$20,000$ is $83$%, then $\frac{20000}{83}$ is $1$% of the original. Since $X$ is the original, then $\frac{X}{100}$ is also $1$% of the original. That means that the two quantities are equal, so we have:
$$\frac{20,000}{83} = \frac{X}{100}.$$
From here it is easy to solve for $X$: Edited:
Since both sides are equal, if you do the same thing to both sides you will get equal quantities. Or, intuitively, since each of these quantities is 1% of what we want, multiplying them by 100 will give us 100% (which is what we want). So we have
$$100\left(\frac{20,000}{83}\right) = 100\left(\frac{X}{100}\right).$$
Now, on the right hand side, the $100$ cancels with the $100$ on the denominator (which again, makes sense: remember that $X$ was the total, $\frac{X}{100}$ was 1% of $X$, so a hundred times 1% of $X$ gives us back $X$), so we are left with
$$\frac{(100)(20,000)}{83} = X.$$
In general, the Rule of Three works because you have proportionality: if $a$ is to $b$ like $x$ is to $c$, (if $\$20,000$ is to $83$% like $X$ is to $100$%)
$$\begin{array}{ccc}
a & \longrightarrow &b\\\
x & \longrightarrow &c
\end{array}$$
then that means that
$$\frac{a}{b} = \frac{x}{c}.$$
which is solved by multiplying through by $c$, which cancels the $c$ in the denominator on the right hand side; it comes out to exactly the same as the Rule of Three says: multiply across the diagonal of known quantities (in this case, $a$ and $c$) and divide by the remaining one (in this case, $b$). The point of the Rule of Three is to do it by rote and not having to think about setting everything up carefully to get the answer.
It also works if your unknown were the percentage: if $\$20,000$ is $87$%, then what percentage of the original is $\$15,575$? (I just made that number up) Well, using the logic from above, since $\frac{20,000}{87}$ is $1$%, and $\frac{15,575}{p}$ is $1$% (where $p$ is the percentage we are looking for) then we have
$$\frac{20,000}{87} = \frac{15,575}{p}$$
which solving for $p$ by first multiplying across by $p$, then multiplying across by $87$, and finally dividing through by $20,000$ (aka "cross-multiplying"):
\begin{align}
\frac{20,000}{87} &= \frac{15,575}{p}\\
\frac{20,000p}{87} &= \frac{15,575p}{p}\\
\frac{20,000p}{87} &= 15,575\\
\frac{20,000p(87)}{87} &= (15,575)(87)\\
20,000p &= (15,575)(87)\\
\frac{20,000p}{20,000} &= \frac{(15,575)(87)}{20,000}\\
p &= \frac{(15,575)(87)}{20,000} \approx 67.75
\end{align}
so the percentage here is approximately $67.75$%. Using the Rule of Three, you have that $20,000$ is to $87$ like $15,575$ is to $p$, so
$$\begin{array}{ccc}
20,000 &\longrightarrow & 87\\\
15,575 & \longrightarrow & p
\end{array}$$
so to get $p$, you multiply across the known diagonal ($15,575$ times $87$) and divide by the remaining quantity ($20,000$), same operation as before.
Added: Again, intuitively, what we are doing is noting that if $20,000$ is 87%, then $\frac{20,000}{87}$ is 1% of the total; we want to know the $p$ such that $15,575$ is $p$%; whatever it is, $\frac{15,575}{p}$ will also be $1$%, so the two quantities are equal. Multiplying both sides by $p$ makes both sides equal to $p$%. of the total. Multiplying both sides by 87 makes both sides $87p$&%. Since $20,000$ is $87$%, dividing both sides by $20,000$ will give $p$, the percentage we are looking for (since we have $87p$% divided by $87$%).
(But it's important that the Rule of Three only works when you do have proportionality. It works for percentages, it works for linear aggregates, but it does not work for things like exponential growth or decay, because then you don't have proportionality between growth and time elapsed.)