Whose 'words' is the Pasuk referring to?
Targum Yonasan: It also refers to the quarrel they had had previously, where Yishmael claimed that he Avraham's rightful heir, since he was the firstborn, whereas Yitzchak countered that he was the son of Sarah, whereas Yishmael was the son of his mother's maidservant.
Rashbam: After he prided himself with the birth of his son, to make a covenant with Avimelech, despite the fact that the P'lishtim were part of the Cana'ani nations, whom Yisrael would be obligated to wipe out 5 ...
Tana d'Vei Eliyahu (Sof Perek 7): It was due to making a Bris with a Nochri (Avimelech), which is akin to serving idolatry. 6
Riva: Rashi needed to give reasons, for "Achar" always refers to something recent (as opposed to Acharei; refer to 15:1:1:1*), and this was many years after the last matter mentioned.
To which Hashem replied that, he had made the parties in honor of his son, and He would now prove that, if He asked him to give him his son, he would not refuse.
Whereas Yitzchak was only eight days old, and who knows whether had he been asked, he would not have objected.
To which Yitzchak replied that, Yishmael was boasting about the one limb with which he had been Moser Nefesh, whereas he (Yitzchak), who was now thirty-six, would gladly give Hashem his whole body, if he was asked to do so.
Lev Eliyahu (Bereishis p.62): Even though Avraham's motives were pure, for the sake of Shalom, that Hashem testified about him, it caused Nochrim to rule over Yisrael.
What was the significance of the test to which Hashem now put Avraham?
Rashbam: ... He made life difficult for Avraham, as if to say, 'Bring your son as an Olah on the Mizbe'ach and see whether the covenant you made with Avimelech will help you!' 1
Rashi and Targum Yonasan (inferred): To counter the arguments of the Satan and of Yishmael. 2
Ramban: The concept of Nisayon, based on man's free-will to do or not to do, is to turn man's potential into practice, so as to reward him for his actions, and not just for his thoughts. 3
Seforno: To turn Avraham's love and fear of Hashem from theory into practice, as the purpose of the creation of man is to emulate His Creator as far as is humanly possible.
See Rashbam, who brings many examples, to prove that whenever the Torah writes "Achar ha'Devarim ha'Eileh", it refers directly to the preceding Parshah (See for example, Bereishis 15:1). Refer also to 22:1:1:4.
The Ramban adds that Hashem only tests Tzadikim, who He knows will pass the test and earn the reward that He has in store for him (as the Pasuk says in Tehilim, 11:8 ).
Hadar Zekenim (verse 14): Avraham said "Hashem, you know what is in man's heart - you need not test people!" Hashem said, I tested you so I can show others how you fear Me.
According to this explanation, 'Nisah' is a Lashon of 'Neis' (a flagpole, as in 've'Sa Neis le'Kabetz Galuyoseinu' - in the Amidah)
Why does it say that Hashem tested Avraham, and not Yitzchak?
Alshich: Avraham needed to act contrarily to his primary Midah (Chesed). Yitzchak's Midah (Din) was to be tough (if needed) against everyone in order to fulfill Hashem's word. Refer to 22:12:2:5.
Machatzis ha'Shekel (OC 591:7, b'Sof): The Akeidah was a greater test for Avraham, for he would be pained constantly afterwards over the loss of his son. 1 Yitzchak would be pained only for a short time. 2