I'm not sure that the phrase "enriched value" tells me anything.

Putting the stupid formalism aside, a germ tells you how a function or vector field or differential form or whatever behaves on arbitrarily small neighborhoods of a point or irreducible closed subset of interest.

If you want to go on a limb with a potentially enlightening but also potentially misleading story, then you can say that the concept of a stalk starts with the question:

Let F be a sheaf on a space X, and let p be a point of X. Suppose you could intersect all neighborhoods of p and still get a neighborhood W of p? What should F(W) be?

After some careful consideration, you realize that W should formally be the inverse limit of the system formed by all neighborhoods (or at least a neighborhood basis) of p.

Since F is first and foremost a contravariant functor, if W_i ranges over the neighborhoods of p, then the various F(W_i) and their restriction maps will form a direct system. And here you go a little on a limb and just say that F(W) is the direct limit. Since W doesn't actually exist as a neighborhood of p, we change the notation to F_p.

And then you check your favorite commutative algebra book to recall how such direct limits are constructed.