Perhaps this will help. This answer is assuming that step refers to the step length.
You are correct that x_new and x_old are measured in the input space, and f(x) is measure in the output space. However, we are considering $\frac{df}{dx}$ and not $f$, so let's look closer at $\frac{df}{dx}$.
Just like f(x), $\frac{df}{dx}$ is a function on the input space. Therefore, let me write it as $\frac{df}{dx}(w)$, where $w$ is some point in the input space. Intuitively, if we are currently at $w$ then $\frac{df}{dx}(w)$ is the direction in the input space which $f$ increases the most. More precisely, $\frac{df}{dx}(w)$ is the gradient of $f$ at $w$. As an example, consider $f:\mathbb{R}^2\to \mathbb{R}$ defined by $$f\begin{pmatrix}x\\y\end{pmatrix}= x^2+y.$$
The gradient is
$$\frac{df}{d_{x,y}}\begin{pmatrix}w\\z\end{pmatrix}= \begin{pmatrix}2w\\1\end{pmatrix}.$$
So this says that at point $\begin{pmatrix}w\\z\end{pmatrix}$, the direction that provides the steepest ascent is $\begin{pmatrix}2w\\1\end{pmatrix}$. For example, if we are currently at the iterate $x_{old} = \begin{pmatrix}0\\0\end{pmatrix}$, then the direction we want to steer our algorithm if we want to ascend the quickest is
$$\frac{df}{d_{x,y}}\begin{pmatrix}w\\z\end{pmatrix} = \begin{pmatrix}0\\1\end{pmatrix}.$$
Therefore, back to your question, the update formula reads like this
x_old and x_new are points in the input space
df/dx (better written as $df/dx(x_old)$) is a direction in the input space
step is just a scalar telling you the step length to step in a certain direction.
