That is, that we sense that there is some ineffable object to our desires and that this proves that there is something beyond this World and life (for it makes the enjoyment of those goods pale in comparison). He used this, in one instance, as proof of God.
As pointed out in the comments, he was primarily basing it on his own experience. But he wasn't the first to do so. For instance, Augustine opens his Confessions by praying to God that, "Our hearts are restless until they find rest in you."
There aren't a whole lot of explicit statements of this in the Bible, but it seems assumed in a lot of places. Jeremiah inveighs against Israel's idolatry by telling them that God alone satisfies: "My people have committed two sins: They have forsaken me, the spring of living water, and have dug their own cisterns, broken cisterns that cannot hold water." (Jer 2:13) The psalmist draws a parallel between how his soul longs for God and how animals pant for water. (Ps 42) Again, the idea is searching for satisfaction, and not finding joy until it is found in God.
In Acts 17, Paul goes to Athens to preach and is stunned at the amount of idols. But one thing in particular catches his attention: an altar "to an unknown God." He tells them that the unknown God can be known, and uses this occasion to preach the Gospel.
Modern missionary Don Richardson begins his book Eternity in their Hearts with the story of that altar, and uses it as a jumping-off point for how cultures across the globe are restless for an unknown God which they catch glimpses of in nature, and of how such cultures are ripe for harvest when missionaries come to teach them of the rest that Jesus offers.
The book is named for a verse in Ecclesiastes: "[God] has also put eternity in their hearts, but man cannot discover the work God has done from beginning to end." (Ecc 3:11) That brings me to the strongest point. The entire book of Ecclesiastes. From beginning to end it's about a man searching high and low for joy. He tries hedonism, he tries philanthropy, he tries accumulating knowledge, and he comes up empty. After trying on every different philosophy and practice, he concludes one thing: "Fear God and keep his commandments, for this is the whole duty of man." (Ecc 12:13)
The Rolling Stones' (I Can't Get No) Satisfaction is a good modern retelling of that journey, minus (unfortunately) its conclusion. But the conclusion is simple: "Come to Me, all who are weary and heavy-laden, and I will give you rest." (Mt 11:28)