One way to visualize a vector field is to imagine that we have little arrows at each value of x, y and z that point in the direction of v with the number of arrows proportional to the magnitude of v at that point. The sign of the divergence of v at (x, y, z) tells us whether more arrows come in to (x, y, z) than go out or vice versa.
A positive divergence means that (x, y, z) is a source and more arrows go out than come in; negative means that
(x, y, z) is a sink; 0 means that the number of arrows coming in and going out is the same. The magnitude of the divergence is proportional to the magnitude of the imbalance of arrows.
All this follows from the divergence theorem which we shall soon discuss.

Exercise

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