When Yahveh offered the Torah (Law of Moses) to the Israelites, He dictated all the terms of the covenant (the Torah was the Old Covenant). One of the terms was that the Israelites would be "cursed" (אָרוּר) if they did not do everything commanded of them in the Torah.
In Deu. 27:26, it is written,
Cursed is he who does not confirm the words of this law, to do them, and all the people shall say, "Amen."
אָרוּר אֲשֶׁר לֹא יָקִים אֶת דִּבְרֵי הַתּוֹרָה הַזֹּאת לַעֲשׂוֹת אוֹתָם וְאָמַר כָּל הָעָם אָמֵן
Now, in Deu. 27:26, it doesn't say "all the words...," but it is implied, and elsewhere there is explicit mention of all the words of the Torah.
For example, in Exo. 24:7, it is written,
And he took the book of the covenant and read in the ears of the people, and they said, "All that Yahveh said, we will do and be obedient."
וַיִּקַּח סֵפֶר הַבְּרִית וַיִּקְרָא בְּאָזְנֵי הָעָם וַיֹּאמְרוּ כֹּל אֲשֶׁר דִּבֶּר יַהֱוֶה נַעֲשֶׂה וְנִשְׁמָע
Consequently, he who neglects to keep even one commandment is cursed. The apostle Paul is merely reiterating a fact of the matter, and that is that no one can keep the Law perfectly, therefore, everyone under the Old Covenant (under the Law) has subjected himself to a curse.
In Gal. 3:10, it is written,
For as many as are of the works of the law are under a curse, for it is written, "Cursed is everyone who does not continue in all that is written in the book of the law, to do them."
ὅσοι γὰρ ἐξ ἔργων νόμου εἰσὶν ὑπὸ κατάραν εἰσίν γέγραπται γὰρ Ἐπικατάρατος πᾶς ὃς οὐκ ἐμμένει ἐν πᾶσιν τοῖς γεγραμμένοις ἐν τῷ βιβλίῳ τοῦ νόμου τοῦ ποιῆσαι αὐτά
Did this give the Israelites permission to ignore obedience to the commandments of the Torah? No, for they made an oath to keep the commandments. Rather, Yahveh gave provision for the atonement of their sins via Temple sacrifices "until the seed should come to whom the promise was made" (Gal. 3:19), i.e. the Lord Jesus Christ and the New Covenant in his blood.