What Does it Mean for an Extra Dimension to Have Size?

The way we measure length is to use the metric tensor. Any spacetime has a metric tensor associated with it, and it's the metric tensor that is responsible for the notion of distance.

To make this a little less abstract consider a concrete example. In flat spacetime the metric tensor is just the following:

$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $$

Suppose you want to measure the distance between two spacelike separated points $(t, x_0, y_0, z_0)$ and $(t, x_1, y_1, z_1)$ - the two points are simultaneous in our coordinates so $t_0 = t_1$. To measure the distance you draw a straight line between the points and integrate $ds$ along it. In flat space the integral simply gives:

$$ s^2 = (x_1 - x_0)^2 + (y_1 - y_0)^2 + (z_1 - z_0)^2 $$

which you should recognise because it's just Pythagoras' theorem in 3D.

So if you're trying to measure the size of a compact dimension start by setting up your local coordinates then just trace a path parallel to the dimension you want to measure. If the dimension is compact the path will return to your original spatial coordinates, and you measure it's length simply by integrating $ds$ along it.

I'm not an expert on this particular topic, but I believe I can answer your question.

There are different kinds of "dimensions". The standard 3 spatial dimensions we live in are infinite in extent. However, one can also imagine dimensions that have a periodicity (like a circle). In such cases, there is a "size" to the dimension that refers to the distance you can translate before you arrive back at where you started. For example, in a circle if we rotate by $2\pi$ then we arrive back at $0$ so we can think of the circle having a "size" of $2\pi$.

Thus, small dimensions are those that curve back on itself such that the total "length" you can move along the dimensions is "small" (how to quantify small here I am not sure).

From this we now see where the usual ant and telephone wire analogy comes from. The analogy can only be taken so far though, because, as you pointed out, the ant and telephone wire exist in a 3-d space which allows for the curvature and periodicity in the wire. This type of curvature is called extrinsic curvature since the curves are embedded in a larger space. The type of periodic dimensions that are small are intrinsically curved, meaning it is the dimension itself that is curved not objects within it.