The equality of all mankind is not a Jewish idea. It is a Greek/Roman/Western-democratic idea that Jews have assimilated over time. Humanity does not all exist on one level. Let's look at how Rabbi Moshe Chaim Luzzatto, ztz"l explains it with commentary provided by Torah.org's Rabbi Yaakov Feldman...
PART 1, PART 2, PART 3, PART 4
Derech Hashem, Part II, Chapter 4 - Israel and the Nations, Paragraphs 8 and 9
 When the world was divided into seventy nations, God appointed seventy [directing] angels as Officers (Sarim) in charge of these nations, to watch over them and attend all their needs.
Thus, God does not oversee these nations except in a general manner. It is each one's directing angel who takes care of the details, through the power that God gives it for this purpose. God thus told Israel (Amos 3:2), "Only you have I known among all the families of the earth."
This does not mean, however, that the details are withheld from God's knowledge. This cannot be true, since everything in all creation is perceived by God and revealed to Him. What it does mean, however, is that God does not oversee and directly influence their details. This will be further clarified in a later section.
 God thus made the rectification and elevation of all creation totally dependent on the Jews. To the extent that this can be expressed, we can thus say that He subjugated His Providence to them. Through their deeds, they can cause [His Light] to shine forth and have influence, or, on the other hand, hold it back and conceal it.
The deeds of the other nations, on the other hand, do not add to or subtract from the state of creation, nor do they cause God to reveal Himself or withdraw. All they can do is bring about their own gain or loss, and strengthen or weaken their own directing angel.
Although God does not involve Himself expressly in the direction of the nations in detail, it is possible that He should exercise His Providence over them when it is required for the sake of Israel. This may be because of a single Jew, or for a number of them. Such a case, however, falls under the category of means, discussed in the previous chapter.
Torah.org - Rabbi Yaakov Feldman
It's not only our interactions with G-d through the mitzvah system that draws our people especially close to Him. It's His interactions with us as well. We'll explore that idea now, as well as the vital role we Jews play in the great rectification of the universe, as we conclude our explanation of the chosen-ness of the Jewish Nation.
We're taught that G-d interacts with other peoples through the agency of their appointed archangel, as each of the seventy core nations of the world has a particular archangel overseeing its perpetuation and well-being.
The point to be made here is that our people's welfare is *directly* overseen by G-d Himself. After all, our life's mission, as we've indicated, is to draw close to and commune with G-d. In fact, though, the interaction is two-sided; for we commune with G-d in close proximity, and He communes with us in kind.
That's not at all meant to imply that G-d has abandoned or is unaware of other peoples! Who would dare even think that of G-d Almighty? The point is that He interfaces with their archangels directly and it's *they* who directly interact with the nation under their auspices. Understand, though, that there's a very fine and subtle metaphysical difference between direct interaction on G-d's part and indirect, but a real one nonetheless. There are instances though in which G-d does directly interact with non-Jews or an entire Gentile nation, but they're exceptions to the rules, and with a particular end in mind.
Thanks to that direct, dare we say "intimate" interaction between G-d and ourselves, we Jews thus play a crucial role in G-d's ultimate aim: universal rectification. For G-d allows His will to be swayed by our wishes and deeds, in a manner of speaking, the way all lovers succumb to their beloved's wishes (though the truth be known, G-d allowing His will to be swayed is in fact a playing out of His will in the end -- but that's beyond our discussion here).
Thus we have it within us to draw down G-d's light into the entire world or to withhold it, by virtue of the fact that we interact with G-d so broadly and all-encompassingly thanks to His mitzvah-system. Others though, with less mitzvot and hence less direct intervention with G-d, haven't as broad an effect upon the world at large, though they certainly do affect their archangel's being as well as their own with their deeds.