The phrase "advanced in years" in 24:1 also appears in 18:11, which states that Avraham and Sara were old advanced in years, and then 18:11 records that Sara was no longer menstruating. 18:12 continues this theme, as the verse records that Sara stated that both she and Avraham were too old to have children. We see that the phrase "old advanced in age," means not just that the person is old but that with this old age, the person is no longer virile or fertile. Avraham and Sara were in this state prior to the announcement of Yitzhak's birth, and then G-d performed a miracle to restore Avraham's virility and Sara's fertility, 18:14. What happened since then? It appears that the normal course of time again returned, and they did not have any more children after Yitzhak.
Accordingly, the first half of 24:1 is stating that Avraham was no longer virile, and then the blessing of everything was that G-d returned to Avraham his virility. Maybe G-d did this as a comfort to Avraham after the death of Sara, that with his renewed strength, he could re-marry. However, before he re-married, he wanted Yitzhak to be married since many times it is difficult for children when their parents re-marry. To minimize any problems, Avraham waited to re-marry until after Yitzhak had found a wife. Furthermore, as Yitzhak was still at least 37 and not married, he decided to move things along by appointing a servant to find a wife for Yitzhak. After, the servant's successful mission, Avraham re-marries, 25:1, and and with his renewed virility he had more children, 25:2. This renewed virility and children might be the beginning of the fulfillment of the blessing to Avraham at the akedah, 22:17.
This understanding of 24:1 can answer another question. 25:2 records that Avraham had six sons with Ketura, which is quite surprising. Sara died when Avraham was 137, and already when he was 99, both he and Sara had expressed doubts that he was able to father any children, 17:17, and 18:12.
Netziv (on 25:2) explains that once Avraham was given the ability to father Yitzhak at the age of 100, then he never lost this ability. This is difficult since then we would have thought that Avraham and Sara would have had more than one child.
Hoffmann (1969) argues that although people lived longer in those days, the years were proportional to modern day lives. Thus, he claims that when Avraham was 140 this was comparable to being 56 in modern times, which allowed him to have had children without requiring any miracles. Hoffman then argues that the miracle of Yitzhak’s birth was that Sara was able to give birth. He follows Ramban’s explanation of 17:17, that Avraham was expressing doubt that he could have a child with Sara since they had not been able to do so before, but not that he himself could not father a child. While it must be that the aging process for Avraham and Sara was different from modern times, as how else could they live such long lives (see our discussion, "Longevity and the aging process in the Torah"), yet still 18:12 remains unexplained, as Sara says that Avraham was too old when he was 99. (Presumably, Ramban and Hoffmann understand 18:12 as they did 17:17, but it is a difficult explanation.)
Luzzatto (on 25:1) writes that there is no doubt that Avraham married Ketura before Sara died. This would seem to explain how Avraham was able to father the children of Ketura, but it seems difficult to believe that Sara would have allowed Avraham to take another wife after all the fights she had with Hagar. Also, presumably Avraham did not marry Ketura before he married Hagar, since then there would have been no need to marry Hagar once Avraham had fathered children with Ketura.
A simpler explanation on 25:2 is our explanation of 24:1 that G-d restored Avraham's virility to him after Sara died. This strength remained with him for enough years that he was able to father the six sons recorded in 25:2 and possibly more depending on how one understands 25:6.
Hoffmann, David Tzvi (1843-1921), 1969, Commentary on Genesis, Bnei Brak: Nezach.