Note that the tangent plane does not necessarily touch the manifold at just one point (consider the tangent plane to a cylinder, or to a plane itself). The definition is actually very close to what I would consider intuitive for "tangent": the tangent plane at a point $p$ on a manifold passes through $p$, and for every curve $\gamma$ on the manifold through $p$, the tangent line to $\gamma$ at $p$ lies within the tangent plane...
Here's another way to look at it: suppose you choose coordinates for your ambient space so that the origin is at $p$, and the $x$ and $y$ directions span the tangent plane, and now locally parameterize the surface by a height map $z(x,y)$ on the tangent plane. Then $z$ has a critical point at $(0,0)$. To see this, notice that for any vector $(a,b)$ in the tangent plane, the curve $\gamma(t) = z(at, bt)$ must have velocity tangent to the plane at $(0,0)$, so that
$$\nabla z \cdot (a,b) = 0.$$
The case where the critical point is a strict local minimum or maximum is what you usually think of as a "just touching" tangent plane, but notice that $z$ could have a saddle point instead, or there could be a whole line or patch containing $(0,0)$ consisting of critical points (as in the case of the tangent plane to a cylinder.)