Your confusion is understandable (I was experiencing almost the exact same problem recently). What I seem to gather from you is that, you see how a traditional tangent vector V can be thought of as a derivation (by the act of taking the directional derivative through V), but you fail to see how an abstract derivation can necessarily be thought of as a traditional tangent vector.
This is understandable... I don't find it at all obvious that an arbitrary derivation at a point p is always basically the taking of a directional derivative. That this is in fact the case is probably best embodied in the following theorem:
If a is in R^n, let Dx1[a], ... , Dxn[a] denote the derivations defined by taking the partial derivative of f(x1, ..., xn) with respect to the variables x1, ..., xn. Then ANY derivation at the point a can be written as a unique linear combination of these derivations. That is, Dx1[a], ... , Dxn[a] form a basis for tangent space of all derivations at a on the manifold R^n.
Again, I don't find this obvious. But once you've convinced yourself of it (I'm sure your book has the proof), the rest follows fairly easily. We can identify the basis for our "traditional" tangent vectors with corresponding derivations. For example, the vector V = (1, 0, 0, ..., 0) is identified with Dx1[a] (convince yourself that taking the directional derivative of of f with respect to the vector V is precisely the same as taking the partial derivative of f with respect to x1).
Of course, everything I've said so far only goes for the rather boring manifold R^n. But really, it goes for any smooth manifold, since we are ultimately identifying open subsets of the manifold with R^n anyway. When we deal with a function f : M --> R, we are probably just going to write it in terms of its coordinate representation anyway, i.e., to think of it as a map f : R^n --> R. And in that sense we are quite justified in carrying over our previous results about R^n into an arbitrary manifold M.
Finally, keep in mind I am no expert in Differential Geometry, so if I've contradicted anything you might read in your book, you should probably believe it over me. In any case, I hope it helped, let me know if it did (or didn't). Take care