Yosef and Binyamin saw before them the long-term, historic implications of what is happening around them.
After Yosef reveals his identity to his brothers, he reassures them that he harbors no hard feelings towards them for their mistreatment of him, and then embraces them and weeps. The Torah places particular emphasis on his embrace and with his only full brother, Binyamin, mentioning that the two brothers embraced another and cried on each other's shoulder (see 45:14). Rashi, citing the Gemara in Masekhet Megila (16b), comments that Yosef and Binyamin wept because they foresaw the calamities that will befall their descendants in each of their territory. The Sanctuary of Shilo was situated in the territory of Efrayim - Yosef's son - and ultimately destroyed. Similarly, the two Batei Mikdash, built in the region of Binyamin, were both burned to the ground.
Why would this prophetic vision of these calamities arise particularly now, during this emotional reunion between Yosef and Binyamin? Why at this moment would they bring to mind the great tragedies that would later befall the Jewish people?
On one level, this Gemara perhaps emphasizes the brothers' awareness that this entire ordeal marked the beginning of the Egyptian exile. As heartwarming as this family reunion was, the twelve brothers understood that the prophecy to Avraham Avinu of "Your offspring will be a foreigner in a land not their own" (Bereishit 15:13) was unfolding before their very eyes. Chazal saw this exile as the prototype of the tragic exiles Am Yisrael will experience much later in history. Yosef and Binyamin lament the pain and suffering that will unfortunately befall their descendants until they are finally reunited in their ancestral homeland - rather in the foreign, hostile land of Egypt. This is the approach taken by the Yefei Toar on Parashat Vayigash.
We may also suggest a more specific association between the three calamities mentioned by the Gemara and the narrative in Parashat Vayigash. Yosef and Binyamin lament the "sin'at achim," fraternal strife, that will ultimately spell disaster for the Jewish people and bring about their downfall. They look at the current situation that has emerged, the absurdity of ten brothers conspiring to eliminate the eleventh, and weep. Although the wounds of this particular chapter of hatred appear to have mended (to one extent or another), Binyamin and Yosef foresee this story's sorrowful repetition throughout Jewish history. Most famously, Chazal (Masekhet Yoma 9b) attribute the destruction of the Second Temple to sin'at chinam - baseless hatred among Am Yisrael.
There are indications that variations of sin'at chinam led to the other two calamities, as well. In describing the events leading up to the destruction of Shilo, Sefer Shemuel I (chapter 2) tells of the corruption of the kohanim and the high priest's refusal (or inability) to admonish his sons for their misconduct. The kohanim manipulated those who came to bring sacrifices to the Mishkan and took for themselves meat that was not intended for them. This misuse of the priesthood to cheat others led to the destruction of Shilo at the hands of the Pelishtim.
As for the First Temple, the aforementioned Gemara in Masekhet Yoma attributes this tragedy to Benei Yisrael's violation of the three grave sins of idolatry, murder and adultery. Elsewhere, Chazal (Shabbat 119b; Eikha Rabba 1:20) comment that the destruction occurred because Benei Yisrael did not admonish one another. If open lines of communication had existed, had people trusted and respected one another, then perhaps the nation could have formed a healthy society in which people were open to criticism and would accept the counsel and admonition of their peers. But just as Yosef's brothers looked upon him with suspicion and contempt, and did not consider discussing with him their concerns (which were, presumably, entirely legitimate), so were Benei Yisrael at the time of the Temple unable to develop a level of trust and mutual respect that would allow for healthy communication and honest, constructive criticism.
In conclusion, we should point to one broad message the clearly emerges from Chazal's comment on this verse. At this moment of personal emotion and family reunion, Yosef and Binyamin saw before them the long-term, historic implications of what is happening around them. They managed to transcend the limited realm of the personal and appreciate the repercussions of this ordeal that would manifest only centuries later. This perhaps teaches us an important lesson about broadening our perspective, of having the vision and foresight to concern ourselves with the broader implications of what we do. We should not think only in terms of our immediate, personal contexts, but rather see ourselves and what we do in terms of the eternal, historic destiny and responsibility of Am Yisrael.