The verses immediately before this (vv. 18ff) describe segments of Egyptian society recognizing Hashem and coming to serve Him: "there will be five cities in the land of Egypt... which swear by G-d's name... there shall be an altar to G-d within the land of Egypt... for they will have cried out to G-d because of their oppressors, and He will send them a great person who saves them... and Egypt will know G-d on that day."
Most of the classic commentaries refer this prophecy to the aftermath of Sennacherib's failure to conquer Jerusalem, due to the miraculous destruction of his army (as described in Isaiah 37:36 and II Kings 19:35). Egypt had also been menaced by the Assyrians, and indeed some Egyptians had been taken as hostages and thus were present to witness the event; the display of G-d's power impressed them so much that they (semi-)converted to Judaism and began to worship Him.
So the verses you quoted follow from that. There will come a time, Yeshayahu is saying, when instead of Egypt and Assyria being rival superpowers, with the Land of Israel a buffer between them (and a frequent target of invasion from both sides), all three countries will join in a pact of friendship, in which they will all serve Hashem together. Israel will be "a third" member of this union, recognized as co-equal in importance with the others, rather than a vassal state as it so often was.
The last verse is variously explained. According to the translation you used, it means that once this happens, Egypt and Assyria will deserve the exalted titles of, respectively, "My people" and "the work of My hands," for their recognition of G-d; nevertheless, the Jewish People will retain their special status as having been "My heritage" all along (Radak, Metzudas David). An alternative understanding is that Egypt and Assyria will declare that they now realize the uniqueness of the Jewish People and their special relationship with G-d - i.e., it should be translated, "Egypt [will declare] 'Blessed is My people,' Assyria [will recognize the Jews as] 'the work of My hands,' and Israel [will be, as it has always been] My heritage" (Targum Yonasan, Rashi, Malbim).
I haven't found anyone who says that this refers to events that are to precede Moshiach's coming. However, since as far as I know the situation described in vv. 23-25 has never yet happened, I think it's a logical inference that at least some of this chapter is so. Whether what's happening now is part of that, or a prelude to it - only Hashem knows. We can speculate, but ultimately our job is to make sure that if indeed this is a harbinger of Moshiach's arrival - and by rights, we should view everything that happens as so, since "all of the set times [for the redemption] have already gone by" (Sanhedrin 97b) - that it spurs us to prepare ourselves with teshuvah and good deeds, and reaching out to our fellow Jews to do the same.