There is no biblical commandment to cover the head, and the rabbinic sources on the kippah are unusually tentative for something now treated as the visible marker of an observant Jewish man.
The Talmud contains scattered material rather than a rule. One passage praises a sage who would not walk four cubits bareheaded, on the ground that the divine presence is above; another attributes to a mother the practice of covering her son's head so that the fear of heaven should be upon him. These are presented as the pious conduct of particular individuals, not as obligations on everyone.
Later authorities drew different conclusions from the same material. Maimonides treats covering the head as the practice of a scholar. The Shulchan Aruch records the covering as expected, and the Vilna Gaon is reported to have held that there is no strict requirement in law to cover the head, only a custom of piety. The obligation as generally practised today rests on the accumulated weight of custom, which in halakhah can become binding, reinforced by the principle of not adopting the ways of the surrounding nations.
That history is why the Reform movement in nineteenth century Germany could drop the kippah as a non obligatory custom without claiming to abolish a commandment, and why it has substantially returned in Reform and Conservative settings since, now often worn by women as well.
In Israel the cap is read as a detailed statement rather than a general one. The knitted kippah srugah, the black velvet, the large white Breslov cap and the absence of one map onto religious and political positions with considerable precision, and Israelis read the signal without thinking about it.