The fields transform under finite conformal transformations as$^1$
$$
\Phi^a(x') \mapsto {\Phi^a}'(x) = \Omega(x')^\Delta\,D(R(x'))^{\phantom{b}a}_b \,\Phi^b(x')\,. \tag{1}\label{main}
$$
as given in equation $(55)$ of $[1]$. Let's break it down:
- $\Delta$ is the conformal dimension of $\Phi$.
- $\Omega$ is the conformal factor of the transformation.
- $D$ is the spin representation of $\Phi$.
- $R$ is the rotation Jacobian of the transformation
So let's compute these things. The spin and the conformal dimension $\Delta$ are given. The first thing we have to look at is the Jacobian.
$$
\frac{\partial x^{\prime \mu}}{\partial x^\nu} = \Omega(x') R^\mu_{\phantom{\mu}\nu}(x')\,.
$$
This implicitly defines both $\Omega$ and $R$ and it is not ambiguous because we require $R \in \mathrm{SO}(d)$, namely
$$
R^{\mu}_{\phantom{\mu}\nu} \,\eta^{\nu\rho}\,R^{\lambda}_{\phantom{\lambda}\rho} \,\eta_{\lambda\kappa}= R^{\mu}_{\phantom{\mu}\kappa}\,.
$$
You can immediately see that for the Poincaré subgroup of the conformal group $\Omega(x')= 1$, whereas for dilatations $\Omega(x') = \lambda$ and for special conformal transformations
$$
\Omega(x') = \frac{1}{1+2(b\cdot x') + b^2 {x'}^2}\,.\tag{2}\label{omega}
$$
This can be proven with a bit of algebra. Using your definition of SCT
$$
{x'}^\mu = \frac{x^\mu - b^\mu x^2}{1+-2(b\cdot x) + b^2 {x}^2}\,,
$$
one can check
$$
\frac{\partial x^{\prime \mu}}{\partial x^\rho} \eta_{\mu\nu} \frac{\partial x^{\prime \mu}}{\partial x^\lambda} = \frac{\eta_{\rho\lambda}}{(1-2(b\cdot x) + b^2 x^2)^2}\,.
$$
That means that the Jacobian is an orthogonal matrix up to a factor, which is the square root of whatever multiplies $\eta_{\rho\lambda}$. Then we have to re-express that as a function of $x'$. After some algebra again one finds that it suffices to change the sign to the term linear in $b$.
Finally, how does one compute $R$? Well, it's just the Jacobian divided by $\Omega$. In the case of special conformal transformations one has (there might be mistakes, redo it for safety)
$$
R^{\mu}_{\phantom{\mu}\nu} = \delta^\mu_\nu + \frac{2 b_\nu x^\mu - 2 b^\mu (b_\nu x^2+ x_\nu - 2 (b\cdot x) x_\nu) -2b^2 x^\mu x_\nu }{1-2b\cdot x +b^2 x^2}\,,
$$
which, as before, needs to be expressed in terms of $x'$.
If you are interested in $\Phi$ scalar then $D(R) = 1$ and you can just plug \eqref{omega} into \eqref{main} to obtain the transformation. If you want to consider also spinning $\Phi$ then it's not much harder.
For spin $\ell=1$ the $D$ is just the identity, namely
$$
D(R)^{\phantom{\nu}\mu}_\nu = R^{\phantom{\nu}\mu}_\nu\,.
$$
For higher spins one just has to take the product
$$
D(R)^{\phantom{\nu_1\cdots \nu_\ell}\mu_1\cdots \mu_\ell}_{\nu_1\cdots \nu_\ell} = R^{\phantom{\nu_1}\mu_1}_{\nu_1}\cdots R^{\phantom{\nu_\ell}\mu_\ell}_{\nu_\ell}\,.
$$
Again, by plugging these definitions in \eqref{main} you obtain the desired result.
$\;[1]\;\;$TASI Lectures on the Conformal Bootstrap,
David Simmons-Duffin, 1602.07982
$\;{}^1\;\;$The way the transformations are written in the lecture notes linked above differs a bit from Di Francesco Mathieu Sénéchal. The difference is that Di Francesco et al. make an active transformation $x \to x'$ with
$$
\Phi(x) \mapsto \Phi'(x') = \mathcal{F}(\Phi(x))\,,
$$
while David Simmons Duffin makes essentially the inverse transformation $x' \to x$
$$
\Phi(x') \mapsto \Phi'(x) = \mathcal{F}^{-1}(\Phi(x'))\,.
$$
That is why in the above discussion the indices of $R^\mu_{\phantom{\mu}\nu}$ get swapped when passed inside $D$ as $D(R) = R^{\phantom{\nu}\mu}_{\nu}$. And that's also why we get a factor $\lambda^\Delta$ rather than $\lambda^{-\Delta}$ as Di Francesco et al. This is all consistent as long as it is clear what one is doing.
Best Answer
Let $$\begin{align} \overline{\mathbb{R}^{p,q}}~~:=&~~\left\{y\in \mathbb{R}^{p+1,q+1}\backslash\{0\}\mid \eta^{p+1,q+1}(y,y)=0\right\}/\mathbb{R}^{\times} \cr ~~\subseteq &~~ \mathbb{P}_{p+q+1}(\mathbb{R})~~\equiv~~(\mathbb{R}^{p+1,q+1}\backslash\{0\})/\mathbb{R}^{\times}, \cr &\mathbb{R}^{\times}~~\equiv~~\mathbb{R}\backslash\{0\}, \end{align}\tag{1}$$ denote the conformal compactification of $\mathbb{R}^{p,q}$. Topologically, $$ \overline{\mathbb{R}^{p,q}}~\cong~(\mathbb{S}^p\times \mathbb{S}^q)/\mathbb{Z}_2 .\tag{2} $$ The $\mathbb{Z}_2$-action in eq. (2) identifies points related via a simultaneous antipode-swap on the spatial and the temporal sphere. The embedding $\imath: \mathbb{R}^{p,q}\hookrightarrow \overline{\mathbb{R}^{p,q}}$ is given by $$\imath(x)~:=~\left(1-\eta^{p,q}(x,x): ~2x:~ 1+\eta^{p,q}(x,x)\right). \tag{3} $$ Let $n:=p+q$. [If $n=1$, then any transformation is automatically a conformal transformation, so let's assume $n\geq 2$. If $p=0$ or $q=0$ then the conformal compactification $\overline{\mathbb{R}^{p,q}}~\cong~\mathbb{S}^n$ is an $n$-sphere.]
On one hand, there is the (global) conformal group $$ {\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \}\tag{4}$$ consisting of the set globally defined conformal transformations on $\overline{\mathbb{R}^{p,q}}$. This is a $\frac{(n+1)(n+2)}{2}$ dimensional Lie group. The quotients in eqs. (2) & (4) are remnants from the projective space (1).
The connected component that contains the identity element is $$ {\rm Conf}_0(p,q)~\cong~\left\{\begin{array}{ll} SO^+(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \} &\text{if both $p$ and $q$ are odd},\cr SO^+(p\!+\!1,q\!+\!1) &\text{if $p$ or $q$ are even}.\end{array}\right.\tag{5}$$ The two cases in eq. (5) correspond to whether $-{\bf 1}\in SO^+(p\!+\!1,q\!+\!1)$ or not, respectively. The global conformal group ${\rm Conf}(p,q)$ has 4 connected components if both $p$ and $q$ are odd, and 2 connected components if $p$ or $q$ are even. The (global) conformal algebra $$ {\rm conf}(p,q)~\cong~so(p\!+\!1,q\!+\!1)\tag{6}$$ is the corresponding $\frac{(n+1)(n+2)}{2}$ dimensional Lie algebra. Dimension-wise, the Lie algebra breaks down into $n$ translations, $\frac{n(n-1)}{2}$ rotations, $1$ dilatation, and $n$ special conformal transformations.
On the other hand, there is the local conformal groupoid $$ {\rm LocConf}(p,q)~=~\underbrace{{\rm LocConf}_+(p,q)}_{\text{orientation-preserving}} ~\cup~ \underbrace{{\rm LocConf}_-(p,q)}_{\text{orientation-reversing}} \tag{7}$$ consisting of locally defined conformal transformations. Let us denote the connected component that contains the identity element $${\rm LocConf}_0(p,q)~\subseteq~{\rm LocConf}_+(p,q). \tag{8}$$ The local conformal algebroid $$ {\rm locconf}(p,q)~=~{\rm LocConfKillVect}(\overline{\mathbb{R}^{p,q}})\tag{9}$$ consists of locally defined conformal Killing vector fields, i.e. generators of conformal transformations.
For $n\geq 3$, (the pseudo-Riemannian generalization of) Liouville's rigidity theorem states that all local conformal transformations can be extended to global conformal transformations, cf. e.g. this & this Phys.SE posts. Thus the local conformal transformations are only interesting for $n=2$.
For the 1+1D Minkowski plane we consider light-cone coordinates $x^{\pm}\in \mathbb{S}$, cf. e.g. this Phys.SE post. The locally defined orientation-preserving conformal transformations are products of 2 monotonically increasing (decreasing) diffeomorphisms on the circle $\mathbb{S}^1$ $$\begin{align} {\rm LocConf}_+(1,1)~=~& {\rm LocDiff}^+(\mathbb{S}^1)~\times~ {\rm LocDiff}^+(\mathbb{S}^1) \cr &~\cup~ {\rm LocDiff}^-(\mathbb{S}^1)~\times~ {\rm LocDiff}^-(\mathbb{S}^1),\cr {\rm LocConf}_0(1,1)~=~& {\rm LocDiff}^+(\mathbb{S}^1)~\times~ {\rm LocDiff}^+(\mathbb{S}^1).\end{align}\tag{10}$$ A orientation-reversing transformation is just a orientation-preserving transformation composed with the map $(x^+,x^-)\mapsto (x^-,x^+)$. The corresponding local conformal algebra $$ {\rm locconf}(1,1)~=~{\rm Vect}(\mathbb{S}^1)\oplus {\rm Vect}(\mathbb{S}^1) \tag{11}$$ becomes two copies of the real Witt algebra, which is an infinite-dimensional Lie algebra.
For the 2D Euclidean plane $\mathbb{R}^{2}\cong \mathbb{C}$, when we identify $z=x+iy$ and $\bar{z}=x-iy$, then the locally defined orientation-preserving (orientation-reversing) conformal transformations are non-constant holomorphic (anti-holomorphic) maps on the Riemann sphere $\mathbb{S}^2=\mathbb{P}^1(\mathbb{C})$ $$\begin{align} {\rm LocConf}_0(2,0)~=~&{\rm LocConf}_+(2,0)\cr ~=~&{\rm LocHol}(\mathbb{S}^2), \cr\cr {\rm LocConf}_-(2,0)~=~&\overline{{\rm LocHol}(\mathbb{S}^2)} ,\end{align}\tag{12}$$ respectively. An anti-holomorphic map is just a holomorphic map composed with complex conjugation $z\mapsto\bar{z}$. The corresponding local conformal algebroid $$ {\rm locconf}(2,0)~=~{\rm LocHolVect}(\mathbb{S}^2)\tag{13}$$ consists of generators of locally defined holomorphic (sans anti-holomorphic!) maps on $\mathbb{S}^2$. It contains a complex Witt algebra.
References:
M. Schottenloher, Math Intro to CFT, Lecture Notes in Physics 759, 2008; Chapter 1 & 2.
R. Blumenhagen and E. Plauschinn, Intro to CFT, Lecture Notes in Physics 779, 2009; Section 2.1.
P. Ginsparg, Applied Conformal Field Theory, arXiv:hep-th/9108028; Chapter 1 & 2.
J. Slovak, Natural Operator on Conformal manifolds, Habilitation thesis 1993; p.46. A PS file is available here from the author's homepage. (Hat tip: Vit Tucek.)
A.N. Schellekens, CFT lecture notes, 2016.