Rashi to Genesis 15:10 explains that it was the custom of those making a covenant to split an animal into pieces, and to then pass between the pieces.
Rashi also refers us to Jeremiah 34:18-19 where this practice is explicitly mentioned:
וְנָתַתִּי אֶת־הָאֲנָשִׁים הָעֹבְרִים אֶת־בְּרִתִי אֲשֶׁר לֹא־הֵקִימוּ אֶת־דִּבְרֵי הַבְּרִית אֲשֶׁר כָּרְתוּ לְפָנָי הָעֵגֶל אֲשֶׁר כָּרְתוּ לִשְׁנַיִם וַיַּעַבְרוּ בֵּין בְּתָרָיו׃
I will make the men who violated My covenant, who did not fulfill the terms of the covenant which they made before Me, [like] the calf which they cut in two so as to pass between the halves.
Thus, idiomatically one talks of “cutting” a covenant, referencing the old practice of cutting up an animal as a way of binding the parties to the covenant.