This formula is the result of a linear interpolation, which identifies the median under the assumption that data are uniformly distributed within the median class.
To derive the formula, we can note that since $N/2$ is the number
of observations below the median, then $N/2 - F_{m-1}$ is the number of observations that are within the median class and that are below the median ($F_{m-1}$ is the cumulative frequency of the interval below the median class, i.e. of all classes below the median class).
As a result, the fraction $\displaystyle\frac {N/2 - F_{m-1}}{f_m}$ (where $f_m$ is the frequency of the median class) represents the proportion of data values in the median class that are below the median.
Now if we assume that data are uniformly distributed (i.e., equally spaced) within the median class, multiplying the last fraction by $c$ (total width of the median class) we obtain the fraction of median class width corresponding to the position of the median. Adding the result to $L_m$ (lower limit of the median class), we get the final formula $\displaystyle L_m + \left [ \frac { \frac{N}{2} - F_{m-1} }{f_m} \right ] \cdot c$, which identifies the median.
The following is not a rigorous derivation (a derivation would require a lot of assumptions
about what makes one estimator better than another), but is an attempt to "make sense"
of the formula so that you can more easily remember and use it.
Consider a bar graph with a bar for each of the classes of data.
Then $f_1$ is the height of the bar of the modal class,
$f_0$ is the height of the bar on the left of it,
and $f_2$ is the height of the bar on the right of it.
The quantity $f_1 - f_0$ measures how far the modal class's bar "sticks up" above the
bar on its left. The quantity $f_1 - f_2$ measures how far the modal class's bar "sticks up" above the bar on its right.
Now, observe that
$$
\frac{f_1 - f_0}{2f_1 - f_0 - f_2} + \frac{f_1 - f_2}{2f_1 - f_0 - f_2}
= \frac{f_1 - f_0}{(f_1 - f_0) + (f_1 - f_2)}
+ \frac{f_1 - f_2}{(f_1 - f_0) + (f_1 - f_2)}
= 1
$$
So if we want to divide an interval of width $h$ into two pieces,
where the ratio of sizes of those two pieces is $(f_1 - f_0) : (f_1 - f_2)$,
the first piece will have width $\frac{f_1 - f_0}{2f_1 - f_0 - f_2} h$.
This is what the formula for estimating the mode does. It splits the width of the
modal bar into two pieces whose ratio of widths is $(f_1 - f_0) : (f_1 - f_2)$,
and it says the mode is at the line separating those two pieces,
that is, at a distance $\frac{f_1 - f_0}{2f_1 - f_0 - f_2} h$
from the left edge of that bar, $l$.
If $f_1 - f_0 = f_1 - f_2,$ that is, the modal bar is equally far above the
bars on both its left and right, then this formula estimates the mode right in the
middle of the modal class:
$$
l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} h = l + \frac12 h.
$$
But if height of the bar on the left is closer to the modal bar's height, then the
estimated mode is to the left of the centerline of the modal class.
In the extreme case where the bar on the left is exactly the height of the modal bar,
and both are taller than the bar on the right,
that is, when $f_1 - f_0 = 0$ but $f_1 - f_2 > 0$,
the formula estimates the mode at $l$ exactly, that is, at the left edge of the modal bar.
In the other extreme case, where the bar on the left is shorter but the bar
on the right is the same height as the modal bar ($f_1 - f_0 > 0$ but $f_1 - f_2 = 0$),
the formula estimates the mode at $l + h$, that is, at the right edge of the modal bar.
Best Answer
For median, it follows more or less directly from the assumption that within each group, the data are entirely evenly distributed. Basically the same assumption leads you to the calculation for the mean, so it shouldn't be too much of a stretch.
I don't really know about mode. I don't even know whether it makes that much sense to talk about mode for grouped data. With mean and median, you can at least have a decent idea of which group it's in. Not so much with mode. And I'm not entirely certain about the formula. But it does seem to choose a mode closer to the preceeding class if $f_0$ and $f_1$ are close, and a mode closer to the succeeding class if $f_1$ and $f_2$ are closer. Which is reasonable.