Noether's theorem: meaning of transformation of coordinates

Classical Lagrangian field theory deals with fields $\phi: M \to N$, where $M$ is spacetime and $N$ is the target-space of the fields. We shall for convenience call $M$ and $N$ the horizontal and the vertical space, respectively. OP is in this terminology essentially asking

Q: What is the meaning of horizontal transformations?

A: It is a (horizontal) flow in spacetime $M$. Infinitesimally, it is generated by a (horizontal) vector field $X\in\Gamma(M)$.

Q: How can the horizontal/spacetime coordinates be important if they are just dummy variables in the action $S=\int_{\Omega} \!d^4x~{\cal L}$?

A: Well, as Phoenix87 points out in his answer, there can be a flow in and out of the integration region $\Omega\subseteq M$ which may create boundary contributions. Moreover, $\Omega$ is often considered to be an arbitrary integration region.

Already Noether herself considered both horizontal and vertical transformations in her seminal 1918 paper. There are many examples on Phys.SE where horizontal transformations play a role. See e.g. this and this Phys.SE posts.

When you integrate the Lagrangian density over a certain region $\Omega$, this is in principle allowed to change and this gives you a "boundary" term in the variation. This is well discussed in, e.g., the book of Goldstein (3rd edition), where the correct proof of the Noether theorem is given.