It is an expression denoting lack of substance, vanity and emptiness. The 10 northern tribes (Ephraim) were looking for protection from the kings of both Assyria and Egypt. Carrying oil to Egypt was a gift to the king for consideration of a treaty with them. They were covering their bets, playing both sides against each other. They forgot to rely upon God's strength for protection, but instead sought help from idolatrous, pagan nations.
Benson Commentary on Hos. 12:1-2,
"Hosea 12:1-2. Ephraim feedeth on wind — Flatters himself with vain, delusive hopes, of receiving effectual support from the alliances which he forms. It is a proverbial expression to signify labour in vain, or pursuing such measures as will bring damage rather than benefit. And followeth the east wind — Pernicious, destructive counsels and courses. The east wind was peculiarly parching and noxious, blasting the fruits of the earth; thence it denotes desolation and destruction. He daily increaseth — Hebrew, ירבה, multiplieth, lies and desolation — Or, falsehood and destruction; so Horsley: that is, in multiplying his falsehood, he multiplies the causes of his own destruction. And they do make a covenant with the Assyrians, and oil is carried into Egypt — Here is an example given of their falsehood, or deceit: while they were in covenant with the Assyrians, having engaged themselves to be tributaries to them, they were secretly and perfidiously seeking to make an alliance with the Egyptians, and for that purpose sent oil as a present to the king of Egypt, endeavouring to persuade him to assist them in shaking off the yoke of the king of Assyria: see the margin. The land of Judah abounded with excellent oil, which was much wanted in Egypt. The Lord hath also a controversy with Judah — Though Hezekiah had abolished idolatry, and restored God’s worship in the temple, 2 Chronicles 29:3; 2 Chronicles 31:1, yet there were much hypocrisy and great corruption in the manners of his subjects; for which God’s judgments are here threatened, and the invasion of Sennacherib was actually inflicted, 2 Kings 18:13, &c. " Source:Biblehub