Here is a mathematical derivation. We use the sign convention $(+,-,-,-)$ for the Minkowski metric $\eta_{\mu\nu}$.
I) First recall the fact that
$SL(2,\mathbb{C})$ is (the double cover of) the restricted Lorentz group $SO^+(1,3;\mathbb{R})$.
This follows partly because:
There is a bijective isometry from the Minkowski space $(\mathbb{R}^{1,3},||\cdot||^2)$ to the space of $2\times2 $ Hermitian matrices $(u(2),\det(\cdot))$,
$$\mathbb{R}^{1,3} ~\cong ~ u(2)
~:=~\{\sigma\in {\rm Mat}_{2\times 2}(\mathbb{C}) \mid \sigma^{\dagger}=\sigma \}
~=~ {\rm span}_{\mathbb{R}} \{\sigma_{\mu} \mid \mu=0,1,2,3\}, $$
$$\mathbb{R}^{1,3}~\ni~\tilde{x}~=~(x^0,x^1,x^2,x^3) \quad\mapsto \quad\sigma~=~x^{\mu}\sigma_{\mu}~\in~ u(2), $$
$$ ||\tilde{x}||^2 ~=~x^{\mu} \eta_{\mu\nu}x^{\nu} ~=~\det(\sigma), \qquad \sigma_{0}~:=~{\bf 1}_{2 \times 2}.\tag{1}$$
There is a group action $\rho: SL(2,\mathbb{C})\times u(2) \to u(2)$ given by
$$g\quad \mapsto\quad\rho(g)\sigma~:= ~g\sigma g^{\dagger},
\qquad g\in SL(2,\mathbb{C}),\qquad\sigma\in u(2), \tag{2}$$
which is length preserving, i.e. $g$ is a pseudo-orthogonal (or Lorentz) transformation.
In other words, there is a Lie group homomorphism
$$\rho: SL(2,\mathbb{C}) \quad\to\quad O(u(2),\mathbb{R})~\cong~ O(1,3;\mathbb{R}) .\tag{3}$$
Since $\rho$ is a continuous map and $SL(2,\mathbb{C})$ is a connected set, the image $\rho(SL(2,\mathbb{C}))$ must again be a connected set. In fact, one may show so there is a surjective Lie group homomorphism$^1$
$$\rho: SL(2,\mathbb{C}) \quad\to\quad SO^+(u(2),\mathbb{R})~\cong~ SO^+(1,3;\mathbb{R}) , $$
$$\rho(\pm {\bf 1}_{2 \times 2})~=~{\bf 1}_{u(2)}.\tag{4}$$
The Lie group $SL(2,\mathbb{C})=\pm e^{sl(2,\mathbb{C})}$ has Lie algebra
$$ sl(2,\mathbb{C})
~=~ \{\tau\in{\rm Mat}_{2\times 2}(\mathbb{C}) \mid {\rm tr}(\tau)~=~0 \}
~=~{\rm span}_{\mathbb{C}} \{\sigma_{i} \mid i=1,2,3\}.\tag{5}$$
The Lie group homomorphism $\rho: SL(2,\mathbb{C}) \to O(u(2),\mathbb{R})$ induces a Lie algebra homomorphism
$$\rho: sl(2,\mathbb{C})\to o(u(2),\mathbb{R})\tag{6}$$
given by
$$ \rho(\tau)\sigma ~=~ \tau \sigma +\sigma \tau^{\dagger},
\qquad \tau\in sl(2,\mathbb{C}),\qquad\sigma\in u(2), $$
$$ \rho(\tau) ~=~ L_{\tau} +R_{\tau^{\dagger}},\tag{7}$$
where we have defined left and right multiplication of $2\times 2$ matrices
$$L_{\sigma}(\tau)~:=~\sigma \tau~=:~ R_{\tau}(\sigma),
\qquad \sigma,\tau ~\in~ {\rm Mat}_{2\times 2}(\mathbb{C}).\tag{8}$$
II) Note that the Lorentz Lie algebra $so(1,3;\mathbb{R}) \cong sl(2,\mathbb{C})$ does not$^2$ contain two perpendicular copies of, say, the real Lie algebra $su(2)$ or $sl(2,\mathbb{R})$. For comparison and completeness, let us mention that for other signatures in $4$ dimensions, one has
$$SO(4;\mathbb{R})~\cong~[SU(2)\times SU(2)]/\mathbb{Z}_2,
\qquad\text{(compact form)}\tag{9}$$
$$SO^+(2,2;\mathbb{R})~\cong~[SL(2,\mathbb{R})\times SL(2,\mathbb{R})]/\mathbb{Z}_2.\qquad\text{(split form)}\tag{10}$$
The compact form (9) has a nice proof using quaternions
$$(\mathbb{R}^4,||\cdot||^2) ~\cong~ (\mathbb{H},|\cdot|^2)\quad\text{and}\quad SU(2)~\cong~ U(1,\mathbb{H}),\tag{11}$$
see also this Math.SE post and this Phys.SE post. The split form (10) uses a bijective isometry
$$(\mathbb{R}^{2,2},||\cdot||^2) ~\cong~({\rm Mat}_{2\times 2}(\mathbb{R}),\det(\cdot)).\tag{12}$$
To decompose Minkowski space into left- and right-handed Weyl spinor representations, one must go to the complexification, i.e. one must use the fact that
$SL(2,\mathbb{C})\times SL(2,\mathbb{C})$ is (the double cover of) the complexified proper Lorentz group $SO(1,3;\mathbb{C})$.
Note that Refs. 1-2 do not discuss complexification$^2$. One can more or less repeat the construction from section I with the real numbers $\mathbb{R}$ replaced by complex numbers $\mathbb{C}$, however with some important caveats.
There is a bijective isometry from the complexified Minkowski space $(\mathbb{C}^{1,3},||\cdot||^2)$ to the space of $2\times2 $ matrices $({\rm Mat}_{2\times 2}(\mathbb{C}),\det(\cdot))$,
$$\mathbb{C}^{1,3} ~\cong ~ {\rm Mat}_{2\times 2}(\mathbb{C})
~=~ {\rm span}_{\mathbb{C}} \{\sigma_{\mu} \mid \mu=0,1,2,3\}, $$
$$ M(1,3;\mathbb{C})~\ni~\tilde{x}~=~(x^0,x^1,x^2,x^3) \quad\mapsto \quad\sigma~=~x^{\mu}\sigma_{\mu}~\in~ {\rm Mat}_{2\times 2}(\mathbb{C}) , $$
$$ ||\tilde{x}||^2 ~=~x^{\mu} \eta_{\mu\nu}x^{\nu} ~=~\det(\sigma).\tag{13}$$
Note that forms are taken to be bilinear rather than sesquilinear.
There is a surjective Lie group homomorphism$^3$
$$\rho: SL(2,\mathbb{C}) \times SL(2,\mathbb{C}) \quad\to\quad
SO({\rm Mat}_{2\times 2}(\mathbb{C}),\mathbb{C})~\cong~ SO(1,3;\mathbb{C})\tag{14}$$
given by
$$(g_L, g_R)\quad \mapsto\quad\rho(g_L, g_R)\sigma~:= ~g_L\sigma g^{\dagger}_R, $$
$$ g_L, g_R\in SL(2,\mathbb{C}),\qquad\sigma~\in~ {\rm Mat}_{2\times 2}(\mathbb{C}).\tag{15} $$
The Lie group
$SL(2,\mathbb{C})\times SL(2,\mathbb{C})$
has Lie algebra $sl(2,\mathbb{C})\oplus sl(2,\mathbb{C})$.
The Lie group homomorphism
$$\rho: SL(2,\mathbb{C})\times SL(2,\mathbb{C})
\quad\to\quad SO({\rm Mat}_{2\times 2}(\mathbb{C}),\mathbb{C})\tag{16}$$
induces a Lie algebra homomorphism
$$\rho: sl(2,\mathbb{C})\oplus sl(2,\mathbb{C})\quad\to\quad
so({\rm Mat}_{2\times 2}(\mathbb{C}),\mathbb{C})\tag{17}$$
given by
$$ \rho(\tau_L\oplus\tau_R)\sigma ~=~ \tau_L \sigma +\sigma \tau^{\dagger}_R,
\qquad \tau_L,\tau_R\in sl(2,\mathbb{C}),\qquad
\sigma\in {\rm Mat}_{2\times 2}(\mathbb{C}), $$
$$ \rho(\tau_L\oplus\tau_R) ~=~ L_{\tau_L} +R_{\tau^{\dagger}_R}.\tag{18}$$
The left action (acting from left on a two-dimensional complex column vector) yields by definition the (left-handed Weyl) spinor representation $(\frac{1}{2},0)$, while the right action (acting from right on a two-dimensional complex row vector) yields by definition the right-handed Weyl/complex conjugate spinor representation $(0,\frac{1}{2})$. The above shows that
The complexified Minkowski space $\mathbb{C}^{1,3}$ is a $(\frac{1}{2},\frac{1}{2})$ representation of the Lie group $SL(2,\mathbb{C}) \times SL(2,\mathbb{C})$, whose action respects the Minkowski metric.
References:
Anthony Zee, Quantum Field Theory in a Nutshell, 1st edition, 2003.
Anthony Zee, Quantum Field Theory in a Nutshell, 2nd edition, 2010.
$^1$ It is easy to check that it is not possible to describe discrete Lorentz transformations, such as, e.g. parity $P$, time-reversal $T$, or $PT$ with a group element $g\in GL(2,\mathbb{C})$ and formula (2).
$^2$ For a laugh, check out the (in several ways) wrong second sentence on p.113 in Ref. 1: "The mathematically sophisticated say that the algebra $SO(3,1)$ is isomorphic to $SU(2)\otimes SU(2)$." The corrected statement would e.g. be "The mathematically sophisticated say that the group $SO(3,1;\mathbb{C})$ is locally isomorphic to $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$." Nevertheless, let me rush to add that Zee's book is overall a very nice book. In Ref. 2, the above sentence is removed, and a subsection called "More on $SO(4)$, $SO(3,1)$, and $SO(2,2)$" is added on page 531-532.
$^3$ It is not possible to mimic an improper Lorentz transformations $\Lambda\in O(1,3;\mathbb{C})$ [i.e. with negative determinant $\det (\Lambda)=-1$] with the help of two matrices $g_L, g_R\in GL(2,\mathbb{C})$ in formula (15); such as, e.g., the spatial parity transformation
$$P:~~(x^0,x^1,x^2,x^3) ~\mapsto~ (x^0,-x^1,-x^2,-x^3).\tag{19}$$
Similarly, the Weyl spinor representations are representations of (the double cover of) $SO(1,3;\mathbb{C})$ but not of (the double cover of) $O(1,3;\mathbb{C})$. E.g. the spatial parity transformation (19) intertwine between left-handed and right-handed Weyl spinor representations.
Best Answer
The confusion here arises because we are not fully analogous to non-relativistic QM here.
Given a (quantum or classical) field $\phi$, we usually specify whether it is a "scalar", "spinor", "tensor", whatever field. This refers to a finite-dimensional representation $\rho_\text{fin}$ of the Lorentz group the field transforms in as an element: $$ \phi \overset{\Lambda}{\mapsto} \rho_\text{fin}(\Lambda)\phi$$ But, simultaneously, the quantum field is an operator on the Hilbert space of the theory, and on the Hilbert space there must exist a unitary representation $U$. More precisely, every component $\phi^\mu$ of the quantum field is an operator, and hence transforms as operators do: $$ \phi^\mu \overset{\Lambda}{\mapsto} U(\Lambda)\phi^\mu U(\Lambda)^\dagger$$ It is now one of the Wightman axioms that $$ U(\Lambda)\phi U(\Lambda)^\dagger = \rho_\text{fin}(\Lambda)\phi$$ or, in components $$ U(\Lambda)\phi^\mu U(\Lambda)^\dagger=\rho_\text{fin}(\Lambda)^\mu_\nu\phi^\nu$$ It is by this assumption that it suffices to give the finite-dimensional representation of the quantum field to also fix the accompanying unitary representation on the infinite-dimensional Hilbert space it is an operator on. The infinite-dimensional representations are characterized by Wigner's classification through their mass and spin/helicity. Since the finite-dimensional representations on the fields are also characterized by spins, the mass (from the kinetic term of the field) and the spin of the field (from its finite-dimensional representation) fix the unitary representation the particles it creates transform in.
All of this is often brushed under the rug because for the Lorentz invariant vacuum $\lvert\Omega\rangle$, we have $$ \phi \lvert \Omega \rangle \overset{\Lambda}{\mapsto} \rho_\text{fin}(\Lambda) \phi \lvert\Omega\rangle$$ so knowing the finite-dimensional representation suffices to know how all states the field creates from the vacuum transform, and since the Fock spaces are entirely build out of such states, this is all the practical knowledge about the unitary representation that is usually needed.