Supposing a body moves in a certain path, the work done on that body is the average force in the direction of the path multiplied by the path length. Work can therefore become infinitesimal in two ways:
Force along the direction of the path becomes arbitrarily small (we'll ignore this one, as it's typically not relevant)
The path length becomes arbitrarily small (this is the relevant one)
So, along a very short path, a very small amount of work is done. We'll call this very small amount of work $dW$. The reason we do calculus is to work with these infinitesimals; in fact, one of the central ideas of calculus stems from the fact that a very large number of very small things (or, by extension, an infinite number of infinitesimal things) makes something that is no longer small (i.e. no longer infinitesimal). The mathematical way of stating this is
$$W = \int dW$$
In context, this means that if you divide a path into a large number of small segments, and add up the work done along each segment, you should get the same result as if you calculated the work done along the whole path. The smaller these segments get, the smaller each segment's contribution is to the total work done. So you can see that the work done along each segment $dW$ tends to zero as the length of the segment tends to zero. But you're adding up a larger and larger number of $dW$'s each time, so the total work done stays the same.
In your particular context, you're examining a particular amount of work done over a particular interval of time. Let's suppose that the object being studied has velocity $v$ at the particular instant you want to take $\frac{dW}{dt}$ at. We'll also assume that velocity is continuous over time (in a practical sense, we'll assume that it's essentially constant for short intervals of time). Since $v=\frac{\Delta x}{\Delta t}$, we can write $\Delta x = v\Delta t$. As $\Delta x$ (the path length we talked about above) get smaller and smaller, the time interval being considered also gets smaller and smaller. In fact, as $\Delta x$ becomes the infinitesimal $dx$, tending to zero, the infinitesimal $dt$ also tends to zero, but the relation $dx = v\, dt$ still holds. in other words, $dx$ and $dt$ tend to zero in such a way that the ratio between them is constant.
Note that $dW = F\, dx$ for infinitesimal path lengths (along which we assume that force is constant). Substituting the above relation, we have that $dW=Fv\, dt$. In other words, we have established a relation between the two infinitesimal quantities $dW$ and $dt$: as $dt$ tends to zero, $dW$ also tends to zero, but in such a way that the ratio between them is constant (namely $Fv$). This is the meaning of the derivative $\frac{dW}{dt}$.
