Suppose there's a 2d vector field:
$F(x,y) = (x,y)$.
Meaning that at the coordinate $(0,0)$ it's a $0$-vector, at $(0,1)$ it's a $(0,1)$ vector and so on.
Here's a picture of it:
If you mathematically calculate the divergence of this vector field, you will get the following scalar field:
$F(x,y) = 2$.
So, at every point the divergence is constant and equals 2.
Now I'm trying to understand what that does physically mean. According to Wikipedia:
In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source at a given point. It is a local measure of its "outgoingness" – the extent to which there are more of the field vectors exiting from an infinitesimal region of space than entering it. <..> The greater the flux of field through a small surface enclosing a given point, the greater the value of divergence at that point.
So, for example, we take a point $(0, 0)$ and draw a small circle around it and see how much of the outward going flux there is:
According to this picture, all the arrows point outwards from the circle, no matter how small the circle is. So divergence at point $(0,0)$ is positive, which checks out with the mathematically calculated value of 2. So, at point $(0,0)$ the vector field acts like a 'source', which makes sense.
Now we take another point $(1,1)$ and also draw a circle around it:
On this image there are arrows entering the circle area and the same amount of arrows leaving the circle area, so the net flux at the point $(1,1)$ should be 0, and the divergence at this point also should be 0, but mathematically calculated value is 2. Why?
The same applies to any other point which isn't $(0, 0)$.
Very simply: The arrows pointing into the circle are smaller (and also slightly fewer) than the arrows pointing out of it. The net pointing of arrows is therefore outward.
(In this case, the size difference in arrows may be the most immediately obvious effect that yields a net outward flow, but the "slightly fewer" part should not be overlooked. It is this "slightly fewer" part that captures the fact that the flow density from a source decreases as you move away from the source. That particles spread out as they move away from their source. That physical fields get weaker as you go further away from their source. Basically, in a field with a point source and otherwise no divergence, and a circular region that doesn't contain the source, because fewer arrows are pointing into the region than out of it, the arrows that point out must be smaller.)
You can think of it as what you have at the origin, plus an overall northeast-ward flow, like a river. The flow is strong enough to "overpower" what would be southwest-pointing flow in that region, so all arrows point mainly in the same direction, but there is still more flowing out than there is flowing in.