Consider a curved 2D surface embedded in 3D space. To find tangent vectors at a point on this surface, we can define a parametric function whose output vector runs through the 2D surface. Then we can take the derivative of the parametric function. The derivative is the tangent vector.
In the above, no scalar function $f$ was needed to define the tangent vector. Why is it that we need an arbitrary scalar function $f$ to define the tangent vectors on an abstract manifold (not embedded in a higher space)? The book I'm reading defines a tangent vector as a function from scalar functions $f$ on the manifold to the real numbers, satisfying stuff like the Leibnitz's rule
EDIT- The definition in the book is : The tangent vector at a point $p$ is a function $v_p: F\rightarrow R$. $F$ is the set of functions $f:M\rightarrow R$, where $M$ is the manifold. The function $v_p(f)$ should satisfy linearity and the leibnitz rule.
That's because as an embedded manifold, there are three natural functions given on the surface $S$: The $x, y, z$ coordinates. We only need three functions, not all the smooth ones, roughly because $dx, dy, dz$ spans the cotangent space. In particular, to specify a tangent vector $v$ at a point $p$, it's enough to know $\langle v, dx\rangle_p, \langle v, dy\rangle_p, \langle v, dz\rangle_p$ which are exactly the three components of the derivative vector if $v$ is described by a parametrized curve around $p$.
If you can find such nice functions on an abstract manifold $M$, you won't need to consider general function on it either. But the defintion remains abstract and general to be flexible and universal. And it's not the only way to define tangent vectors. In fact, in physics leaning books, the tangent vectors are often defined as column vectors attached to charts that change under certain rules with respec to chart transition maps.