What is the connection between the Akeidah and the information that Rivka was born to Besuel?
Rashi: After the Akeidah, Avraham thought to himself what would have happened with HaSh-m's promise of children had Yitzchak been sacrificed. He therefore considered marrying him to one of the children of Aner, Eshkol and Mamrei. 1 That was when Ha'Kadosh-Baruch-Hu informed him of the birth of Rivka - his intended mate. 2
Targum Yonasan: Avraham was informed of the birth of Rivka after the Satan told Sarah that Avraham had Shechted Yitzchak (even though he had only been bound), following which she died of heartbreak. 3
These are the "words" (i.e. thoughts) to which the Pasuk is referring. Also refer to 22:20:1.1.
This is also how the Seforno explains the connection. The other details are of secondary importance.
Also refer to 23:2:4:1. This follows the principle that a Tzadik does not leave the world until another Tzadik is born to take his place (Yoma 32b). See Seforno to Bereishis 23:2.
Assuming HaSh-m told him the news immediately after the Akeidah, why does the Torah write "Acharei"? (The Midrash brings a dispute as to whether 'Achar' means 'right after,' and 'Acharei' means 'a while after' - or the reverse. Rashi writes elsewhere that 'Acharei' means 'a while after;' so why does he write here that Avraham was informed of Rivka's birth right after the Akeidah?)
Chizkuni, Mizrachi: It seems that 'Achar' means immediately (exclusively), whereas 'Acharei' can mean long afterwards too (but not exclusively).
Gur Aryeh: We see at times that 'Acharei' does not mean far off. 1 'Achar' is the opposite of 'Kodem' (e.g. item A is after item B); and 'Acharav' (with a Yud, like 'Acharei') is the opposite of 'Lefanav' (item A is behind, or at the tail end, of B). 2 'Acharei' need not indicate a long span of time, only that the verse wishes to make it a distinct new event; whereas 'Achar' means that the two events were as one. 3
Perhaps this a support for the commentaries who disagree with Rashi, who holds that Rivka was already eleven years old at the time of the Akeidah, and that she married Yitzchak when she was fourteen.
For example, "After the death (Acharei Mos) of the two sons of Aharon" (Vayikra 16:1); and that section was taught to Moshe that very day.
To one point of view, Achar means immediately after, and implies a direct association (e.g. Reuven waits on line 'Achar' Shimon); whereas Acharei can be distant (i.e. Avraham came 'Acharei' Noach). To the second view, 'back' and' front' are properties describing an object itself, so ''Acharei' is immediate; whereas Achar may be far-off.
See Gur Aryeh to Devarim 11:30.
Why does the Torah see fit to tell Avraham all this?
Rashi (to 22:23) and Ramban (Ibid.): The purpose of this Parshah is the statement that Besuel bore Rivka; all the rest is secondary. 1
Ramban: Because Milkah was the daughter of Avraham's brother Haran, and HaSh-m saw fit to inform him that his older brother (Nachor) had been blessed with all these children from the daughter of their deceased brother. 2
Gur Aryeh (to 22:23): If so, why are the other sons of Nachor recorded? Nachor fathered twelve sons - eight from his wife, and four from his concubine; these numbers parallel the twelve tribes of Yisrael (Rashi to 22:20). This comes to highlight how Rivka was the appropriate match for Yitzchak. Also refer to 22:24:1:3.
Refer to 22:20:4:2*: See also Rabbi Chavel's commentary on the Ramban.
Why does the Torah write "Milkah Gam Hi"? What is the close comparison between Sarah and her sister Milkah?
Rashi: The Pasuk is actually comparing the families of Avraham and (his brother-in-law) Nachor, inasmuch as the latter, like the former would eventually do, comprised twelve tribes - eight from the main wife, and four from the concubine.
Ramban and Targum Yonasan: When HaSh-m performed a miracle enabling Sarah to bear children in her old age, He did the same on behalf of Milkah, who also gave birth in her old age. 1
Seforno: "Gam Hi" means that besides the concubine (Re'umah), Milkah too, bore children to Nachor. 2
Because, had she given birth to them in her younger years, Avraham would have been bound to know about it.
It is strange to say this however, since a) It has not yet mentioned Re'umah; and b) It would be more appropriate to say that also Re'umah bore children... (Indeed, the Torah writes "Gam" there as well, and writing it in both places is truly odd.)
QUESTIONS ON RASHI
Rashi writes: "'After these things' - While returning from Har ha'Moriyah [after having almost offered up Yitzchak], Avraham contemplated... 'I should have married him off....'" Mizrachi asks - a) Why not explain simply, that after HaSh-m blessed Avraham with increased offspring (22:17); He told Avraham about Rivka, who would bear them? b) The contemplation that Rashi refers to is not written explicitly.
Gur Aryeh: a) "After these things" implies that after one event happened, a different event ensued. Rashi explains that when Avraham contemplated marrying off Yitzchak to a daughter of Aner, Eshkol or Mamrei, who was not his true mate, HaSh-m informed him that Rivka would be his mate instead. 1 b) It is unnecessary for the Torah to write these thoughts of Avraham explicitly; it is obvious that after the Akeidah, Avraham would contemplate marrying off Yitzchak.
Thus, according to Gur Aryeh, this event comes to undo what came previously. According to Mizrachi, however, the birth of Rivka would be a continuation of the same theme, and the phrase 'Acharei ha'Devarim ha'Eleh' would not fit. (EK)