Looking at Lie algebras as your source does (and using the physics convention for elements of the algebra):
I will modify the regular notation somewhat so as to better fit in with physics standards. For a real matrix Lie algebra with its standard/defining/fundamental representation over a complex vector space given by left multiplication, its complex-conjugate representation is given by
$$\overline{\pi}(X)=-X^{\ast}$$
where the star denotes complex conjugation of the entries. The dual representation is given by
$$ \pi^{d}(X)=-X^{t}$$
If the generators are hermitian, as is the case of the Pauli matrices, you can see these two representations are exactly the same, as $-\sigma^{\ast}=-\sigma^{t}$.
On page 75 of your pdf, they show that $\epsilon (-\sigma_k^{\ast}) \epsilon^{-1}=\sigma_{k}$ for $k=1,2,3$ with
$$ \epsilon=\begin{pmatrix}0&1\\ -1&0\end{pmatrix} $$
This basically states that the fundamental representation of $\mathfrak{su}(2)$ is self-dual, as the dual/complex-conjugate/antifundamental representation acts as the standard/fundamental representation when making the change of basis given by
$$ \epsilon\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}-y\\x\end{pmatrix}$$
(Note: in fact, all representations of $\mathfrak{su}(2)$, and therefore $\text{SU}(2)$ as it is simply-connected, are self-dual; see here for a rather more technical explanation, which at its core uses the above change of basis.)
However, for the case of $\mathfrak{so}(3,1)\simeq\mathfrak{sl}(2,\mathbb{C})$ (the latter viewed as a real six-dimensional Lie algebra), first note from your source's basis $\{\sigma_k,i\sigma_k\}_{k=1,2,3}$ that the generators are no longer hermitian, and therefore the dual and complex-conjugate representations do not coincide. With antifundamental we are then referring to the complex-conjugate representation.
What your notes are saying then is that, having the complex-conjugate representation, we should take a change of basis such that the spacial (hermitian) part of the representation acts as the standard representation. (Note the typo in equation 8.83; $\sigma^{\ast}_{k}$ should be $\sigma_k$). But by doing so, the boost part does not act as in the standard representation; it differs by a sign. Basically: you cannot make a change of basis such that the complex-conjugate representation becomes exactly the standard one, and so it is inequivalent to it.
Note that these two representations are still self dual, since transposition ignores the $i$ factor. For the antihermitian operators we have that
$$-\left(i\sigma_k\right)^t=-i\sigma_k^\ast\neq -\left(i\sigma_k\right)^\ast $$
for $k=1,2,3$ so the above trick of changing the basis still works.