Introduction
It is known that under changes of coordinates different fields transform according to their tensorial nature (scalar, vector, etc.) like$^{[1]}$
\begin{equation} \phi(x)\rightarrow\phi'(x')=\phi(x) \end{equation}
\begin{equation} V_{\mu}(x)\rightarrow V_{\mu}'(x')=\frac{\partial x^\nu}{\partial x'^{\mu}}V_\nu(x) \end{equation}
\begin{equation} g_{\mu \nu}(x)\rightarrow g_{\mu \nu}'(x')=\frac{\partial x^\rho}{\partial x'^{\mu}}\frac{\partial x^\sigma}{\partial x'^{\nu}}g_{\rho \sigma}(x) \end{equation}
and so on. We can write an action for a massive free scalar field that is invariant under any change of coordinates
\begin{equation} S=\int d^Dx \sqrt{|g|(x)}\Big\{g^{\mu \nu}(x)\partial_\mu\phi(x)\partial_\nu\phi(x) -m^2\phi^2(x)\Big\}. \end{equation}
If we apply a coordinate change should transform the metric $g$ and the field $\phi$. And since the variables are dummy indeces we can change them as well to get
\begin{equation} S'=\int d^Dx' \sqrt{|g'|(x')}\Big\{g'^{\mu \nu}(x')\partial'_\mu\phi'(x')\partial'_\nu\phi'(x') -m^2\phi'^2(x')\Big\}. \end{equation}
The differential $d^D x$ and the jacobian $\sqrt{|g|(x)}$ transform in opposite ways so they cancel out. In the same way, the change on the metric cancels out with the change in the derivatives. Lastly, the scalar field itself doesn't change so we end up with the same action. This is the statement that this action is invariant under changes of coordinates.
However, scale transformations
\begin{equation} x^\mu \rightarrow x'^\mu =\lambda x^\mu \end{equation}
are a specific type of coordinate transformation and thus this action is also invariant under those. It is commonly known that theories with a mass term break the scale invariance, so the actual meaning of scale invariance in this context is not the same. According to an answer here Simple conceptual question conformal field theory what we actually mean by scale transformation is a change of coordinates $x'^\mu =\lambda x^\mu$ followed by a Weyl transformation that rescales everything
\begin{equation} \phi(x)\rightarrow \tilde{\phi}(x)=\lambda^{-\Delta}\phi(x) \end{equation}
\begin{equation} g_{\mu \nu}(x)\rightarrow \tilde{g}_{\mu \nu}(x)=\lambda^{2}g_{\rho \sigma}(x) \end{equation}
Where $\lambda^2$ is set to cancel the factor that the coordinate transformation induced on the metric. Now the transformed action is
\begin{align} \tilde{S}'= & \int d^Dx' \sqrt{|\tilde{g}'|(x')}\Big\{\tilde{g}'^{\mu \nu}(x')\partial'_\mu\tilde{\phi}'(x')\partial'_\nu\tilde{\phi}'(x') -m^2\tilde{\phi}'^2(x')\Big\} \\ = & \int d^Dx \sqrt{|g|(x)}\Big\{g^{\mu \nu}(x)\partial_\mu\phi(x)\partial_\nu\phi(x) -\lambda^{-2\Delta}m^2\phi^2(x)\Big\}. \end{align}
The kinetic term is scale invariant if $\Delta=\frac{D-2}{2}$ when we add the Weyl rescaling but now the mass term breaks the symmetry. This action is only scale invariant when the mass is $m=0$.
Conflict with the Conformal Group
On the previous calculation the factor $\lambda^{-\Delta}$ in the transformation of the field happens because of the Weyl Transformation, not because of the conformal transformation (which is just a change of coordinates and thus $\phi'(x')=\phi(x)$). However, in many books we see that the generators of the conformal group are$^{[2]}$
\begin{align} P_{\mu} =&-i\partial_{\mu} \\ D =&-i\big(x^{\mu}\partial_{\mu}+\Delta\big)\\ L_{\mu\nu} =&S_{\mu\nu}+i\big(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}\big)\\ K_{\mu} =&i\Big[x^{2}\partial_{\mu}-2x_{\mu}x^{\nu}\partial_{\nu}-\Delta x_{\mu}-iS_{\mu\nu}x^{\nu}\Big]. \end{align}
Scale transformations ($D$) and Lorentz rotations ($L_{\mu\nu}$) can affect the fields even when we transform both the fields and the coordinates as
\begin{equation} \phi_\mu(x)\rightarrow \phi_\mu'(x')=e^{i\theta^{ab}\big(S_{ab}\big)_{\mu\nu}}\phi_\nu (x) \end{equation}
for rotations and
\begin{equation} \phi_\mu(x)\rightarrow \phi_\mu'(x')=\lambda^{-\Delta}\phi_\mu (x) \end{equation}
for scale transformations. Now, although it is clear that the rotation is compatible with the definition of change of coordinates that we did for vectorial objects, the scale transformations make no sense. How is this last transformation compatible with the fact that conformal transformations are just a change of coordinates? The calculations done in the introduction of the post assumed that conformal transformations (as coordinate transformations) don't change fields, while the Weyl transformation is neccesary to produce the scale transformation for $\phi$. But in this last result of the conformal group, the change of coordinates changes the field on its own.
References:
[1] Simple conceptual question conformal field theory
[2] Section 4.2.1. in Francesco's CFT
