We start with a production function: $f(k,l)$, which tells us what quantity will be produced if we use $k$ units of capital and $l$ units of labour. Suppose that we cange that quantity of capital by $dk$ units and change the quantity of labour by $dl$ units. If this change is chosen so that we stay on the same isoquant then it must be that the total production doesn't change:
$$\frac{\partial f(k,l)}{\partial k}dk+\frac{\partial f(k,l)}{\partial l}dl=0.$$
Rearranging, we get
$$\frac{dk}{dl}=-\frac{\frac{\partial f(k,l)}{\partial l}}{\frac{\partial f(k,l)}{\partial k}}.$$
This is the marginal rate of technical substitution, the slope of the isoquant. It has the same interpretation as any other slope. It means that if I increase labour by one unit then I can decrease capital by $$\frac{\frac{\partial f(k,l)}{\partial l}}{\frac{\partial f(k,l)}{\partial k}}$$ units.
Example Suppose we have a Cobb-Doublas production function: $f(k,l)=k^a l^{1-a}$. We have
$$\frac{\partial f(k,l)}{\partial k}=ak^{a-1}l^{1-a}$$
$$\frac{\partial f(k,l)}{\partial l}=(1-a)k^{a}l^{-a}$$
$$MRTS=-\frac{(1-a)k^{a}l^{-a}}{ak^{a-1}l^{1-a}}=-\frac{1-a}{a}\frac{k}{l}.$$
Now we can plug some numbers in to this example to see how it works. Suppose $a=1/2$, and we currently have $k=4$ and $l=1$. Then, we have
$$MRTS=-\frac{1/2}{1/2}\frac{4}{1}=-4.$$
So, for each unit increase in the quantity of labour employed, we can decrease the quantity of capital employed by 4 units.
This works backwards too, so every unit decrease in $l$ would demand a 4 unit increase in $k$ to keep us on the same isoquant.
