Actually, I have two explanations, depending on what you consider more intuitive.
Interval arithmetic
This one is best explained by example. Remember that a value with a certain number of significant figures is supposed to represent that exact value $\pm 5$ in the first insignificant digit. For example, $12.3$ with three significant figures represents anything in the range $[12.25,12.35]$, and $4.6$ with two significant figures represents $[4.55,4.65]$. If you multiply these two ranges, you can get anything between a minimum of $4.55\times 12.25 = 55.7375$ and a maximum of $4.65\times 12.35 = 57.4275$. So strictly speaking, the result is $[55.7375,57.4275]$.
However, it's a lot more convenient to represent this range as a single number, using the same significant figure convention as before, where the uncertainty is $\pm 5$ in the first insignificant digit. We can't actually do this, though; the size of the range is $1.69$, which is not $5\times 10^n$ for any $n$. The sensible options are either to write $60$, which represents the range $[55,65]$, or $57$, which represents the range $[56.5,57.5]$. But there isn't a straightforward rule that justifies the first choice, without doing interval arithmetic every time. The second choice, on the other hand, can be justified by keeping the lower number of significant figures from either of the original operands. So that's what we do.
General error propagation
For a better justification, significant figure rules are chosen as they are because they're a decent match to the general formula for propagation of uncertainties. That general formula is as follows: if you have some quantity $y$ which is a function of several other variables, $y = f(x_1, x_2, \ldots)$, then the uncertainty in $y$ is given by
$$\sigma_y = \sqrt{\biggl(\frac{\partial y}{\partial x_1}\biggr)^2\sigma_{x_1}^2 + \biggl(\frac{\partial y}{\partial x_2}\biggr)^2\sigma_{x_2}^2 + \cdots}$$
Actually, this formula only applies under certain conditions, but they're very common conditions, and it would take a whole separate answer (or even more) to explain all the subtleties. (See here and here for examples.) So let's just take that formula as a starting point.
If a value $x$ has $n$ significant figures, that means its relative uncertainty $r = \frac{\sigma_x}{\lvert x\rvert}$ is between $(5\times 10^{-n})$, if $x$ is a power of $10$, and $(5\times 10^{-(n+1)})$, if $x$ is just less than a power of $10$ (like $99.99999$). Or within an order of magnitude, anyway. You could argue that maybe the $5$ should be a $3$ or something like that, but that gets into the conditions I already decided to skip discussing.
Multiplication and division
I do this case first because it's easier. If the function is $f(x_1, x_2) = x_1x_2$, the general error propagation formula gives
$$\sigma_y = \sqrt{x_2^2\sigma_{x_1}^2 + x_1^2\sigma_{x_2}^2}$$
Or substituting in $rx$ for the uncertainty, all the $x_1$s and $x_2$s cancel out and we get
$$r_y = \sqrt{r_1^2 + r_2^2}$$
Going through the argument for division, the algebra is a little different but you still get to this same formula for relative uncertainty.
There are three cases to consider here:
- $r_1 \ll r_2$: $r_1$ is basically insignificant and you can write $r_y \approx r_2$
- $r_1 \gg r_2$: $r_2$ is basically insignificant and you can write $r_y \approx r_1$
- $r_1 \approx r_2$: in this case you can approximate them as roughly equal and write $r_y \approx \sqrt{2} r_1$. Depending on the exact relative magnitudes, this factor might be a little different, but both $r_1$ and $2 r_1$ are likely to be around the same order of magnitude, which is all that matters for an uncertainty.
One can make this argument a little more precise, but the point is, the resulting uncertainty $r_y$ is more or less the same as the larger of $r_1$ and $r_2$. But remember, $r$ is related to the number of significant figures: $r \approx 10^{-n}$, meaning that a larger relative uncertainty $r$ corresponds to a smaller number of significant figures. So the final quantity $y$ will typically have the same number of significant figures as the less precise (fewer significant figures) of the two operands $x_1$ and $x_2$.
Addition and subtraction
It's also worth checking that this works for addition and subtraction.
If the function is $f(x_1, x_2) = x_1 \pm x_2$, then the partial derivatives are both equal to $1$. So the error propagation formula gives
$$\sigma_y = \sqrt{\sigma_{x_1}^2 + \sigma_{x_2}^2}$$
Here substituting in the relative uncertainty gives
$$r_y(x_1 \pm x_2) = \sqrt{r_1^2 x_1^2 + r_2^2 x_2^2}$$
but this isn't so useful because things don't cancel out. So let's go back to the previous formula, with the $\sigma$s. This time, the three cases break down as follows:
- $\sigma_1 \ll \sigma_2$: $\sigma_1$ is negligible and you can set $\sigma_y \approx \sigma_2$
- $\sigma_1 \gg \sigma_2$: $\sigma_2$ is negligible, so $\sigma_y \approx \sigma_1$
- $\sigma_1 \approx \sigma_2$: following more or less the same reasoning as before, $\sigma_y \approx \sigma_1$ to within an order of magnitude or so
Now, the actual uncertainty $\sigma$ is related, not to the number of significant figures, but to their position. The larger uncertainty corresponds to the leftmost insignificant figure among the two operands, which is exactly what you use.
