Schur-Weyl Duality ( Classical ) and the Double Commutant reference request
If you're referring to Lemmae 6.22 and 6.23 of Fulton and Harris, I agree that some steps are skipped in there. They assume the reader has some familiarity with results in module theory and semisimple algebras, etc.
However that being said any proof of Schur - Weyl Duality that you are going to find is going to use results from module theory. Two alternative resources for Schur - Weyl Duality that come to mind:
Procesi's Lie groups: An Approach Through Invariant Theory - For the double centraliser theorem, look at chapter 6 which discusses semisimple algebras. For Schur - Weyl duality, look at chapter 9 on tensor symmetry.
Daniel Bump's Lie groups - The part on Schur - Weyl duality can be found in the topics section of Bump's book. However IIRC things are done via the representation ring and is not the "usual" approach to classical Schur - Weyl duality.
It would be best if you could say exactly where in Lemmae 6.22 and 6.23 that you don't understand so that I can elaborate on my answer.
Here's my take on part 2 of 6.22:
We first assume that $U$ is irreducible so that $B = \Bbb{C}$. In this case it will then suffice to prove that $U \otimes_A W \cong Uc$ is one dimensional or zero. Firstly by Artin - Wedderburn we get that $A$ is isomorphic to a direct sum of matrix rings $\bigoplus_{i=1}^r M_{n_i}(D_i)$ over some division ring $D_i$. Since there are no non-trivial finite dimensional division rings over $\Bbb{C}$, we conclude that $A = \bigoplus_{i=1}^r M_{n_i}(\Bbb{C})$. Now by assumption $W = Ac$ is an irreducible left $A$ - module and hence is also a minimal left ideal of $A$. We will identify such a minimal ideal in a direct sum of matrix rings. Recall that an idempotent in a ring $R$ is said to be primitive if it cannot be decomposed as the direct sum of two non-zero orthogonal idempotents.
In a semisimple ring such as $A = \bigoplus_{i=1}^r M_{n_i}(\Bbb{C})$, the primitive idempotents are hence those $r$ - tuples of the form $(0, \ldots, e,\ldots, 0)$ for $e$ a primitive idempotent in $M_{n_i}(\Bbb{C})$ for some $i$. A primitive idempotent in $M_{n_i}(\Bbb{C})$ is just an $n_i \times n_i$ matrix $E_{kk}$ for some $1 \leq k \leq n_i$ with all entries zero except entry $(k,k)$. By Theorem 3.1, Chapter 6 of Procesi's book above every minimal left ideal in $A$ is of the form $M_{n_i}(\Bbb{C})E_{kk}$ with $E_{kk}$ a matrix the form described above. Such a left ideal isomorphic to one that consists of tuples with all entries zero except entry $i$. In this entry, all matrices have only one non-zero column, namely column $k$. Similarly $U$ can be identified with the right ideal of $r$ - tuples which are zero except in factor $j$, and in that factor all are zero except row $l$ say. It now follows that $U \otimes_A W$ will be zero unless $ j = i$, in which case $U \otimes_A W$ is isomorphic to the set of matrices that are all zero except in entry $(l,k)$. Hence $\dim U \otimes_A W \leq 1$ and the proof in this case is complete. In the more general case of (ii), decompose $U = \bigoplus_i U_i^{\oplus n_i}$ into a sum of irreducible right $A$ - modules, so
$$U \otimes_A W \cong \bigoplus_i (U_i \otimes_A W)^{\oplus n_i} \cong (U_i \otimes_A W)^{n_k} \cong \Bbb{C}^{n_k}$$
for some $k$. This is clearly irreducible over $B = \bigoplus_j M_{n_j} (\Bbb{C})$.
Despite of very detailed and excellent answer from BenjaLim, let me also list some reference I used (my background is more symmetric group representation theoretic, rather than Lie theoretic, so this gives an alternate flavour to what BenjaLim suggested)
(1) Goodman-Wallach "Representations and Invariants of Classical Groups"; which I think is the clearest reference I have ever read (advantage: they setup double commutant theory abstractly and develop SW duality from that)
(2) Martin "Schur algebras and representation theory"; there are some steps which he skipped, but can be done as an exercise. The term "Schur-Weyl duality" is never used in the text, and the actual "Proof" is scattered in different parts of the book, though not very hard to spot.
(3) Etingof's course notes "Introduction to Representation Theory", this looks more like a concise version of Goodman-Wallach.
In a way, I should list Green's book "Polynomial Representation of GL_n", which is what my advisor recommended me to read, but if you just want "Schur-Weyl duality" without the appearance of Schur algebra, then this is not recommended, as it will take quite a long way to get to SW duality.
Also see this Math.SE question: Understanding the proof of Schur-Weyl Duality