The sentence is to be taken in the general context of the totally novel significance of the New Testament as opposed to the Old: the Old Testament, the Law and the Prophets were for Paul and the New Testament authors a preparation, a foreshadowing of the New, of Christ (Col. 2:17), who, as the "New Adam - the vivifying spirit”, “the Lord from Heaven” (1 Cor. 15:45-47), brought that ontological change in humanity, defeating and abolishing the consequence of the first, earthly Adam’s sin, which was death (1 Cor. 15:21; 15:21). This was unimaginable before Him, for He conquered death by destroying its very foundation - the sin, and everybody who will be co-crucified with Jesus Christ and die for the temptations of this world, will be resurrected into a new life in Christ, He Himself working in a believer's heart, transforming him to a "new creation" (2 Cor. 15:17), who will no more experience that unhappy conscience of a pre-Christ-ly righteous men, so vividly depicted in Romans, when heart is split between two drives – the desire to follow the Law and the unbeatable proclivity for the opposite (Romans 7:19), but after the advent of Christ righteousness becomes a sweet and light yoke (Matt. 11:30), for not he, a believing man alone, but Christ is working this righteousness through His grace, of course, with a believer’s free response and co-action/synergy (1 Cor. 3:9) with this "grace", which in a more technical theological wording is the same as God's operation in human heart (Col. 1:29).
In this light should be understood Paul's given passage and I see two possible ways of its interpretation:
1) we (Jews) knew Christ before wrongly, as some kind of a political messiah who would restore the political kingdom of Israel and, in our present historical situation, liberate us from Romans and establish something like "Make Israel great again!"; but without transformation of inner core of our fallen nature. But now, after the advent of Jesus Christ, we understood the futility and vanity of our previous crude and earthly expectations and see Him spiritually, through the Spirit of God, through whom we perceive His divinity and Lordship (1 Cor. 12:3), and realize that He came in order to transform us from within and destroy the dominion of sin and death in us, establishing His and Father’s and Holy Spirit’s Kingdom in an invisible way (Luke 17:20) already in this life, to be continued eternally and infinitely after the physical death, which for Christians is already a desirable acquisition (Phil. 1:21).
2) until Christ was on earth, we, the apostles (although not Paul himself) and those who followed Him, knew him more physically and less spiritually, more humanly, than divinely, even few days before He was crucified, we demanded high ministerial posts in His kingdom as if He were just an earthly king (Matt. 20-21); we could not yet contain his sayings (John 16:12) and only after receiving the Spirit would be taught by the Spirit and be able to know spiritual things, i.e. the supra-political and eternal dimension of His Kingdom, which is the same as the Kingdom of the Father. That is why He even said that “it is better for you if I go, for the Spirit will not descend upon you unless I go (John 16:7) and then I will attract you in a new way (John 12:32) and will you start a new way of longing for Me and thinking and weighing all that I have taught you, through the Spirit of Truth, who eternally proceeds (ἐκπορεύεται) from the Father; and then you will know Me even better and in a more elevated way, than while I was physically with you; and, moreover, those who have not seen Me physically are in fact even more blessed than you (John 20:29) for they will pass to the spiritual understanding of my mission, as establishing the eternal Kingdom of God in their hearts, more easily.”
Thus, this is the meaning of also Paul's saying that worldly or carnal understanding of the messiah, of Jesus Christ, has now been supplanted by a lofty, spiritual understanding of His eternal Kingdom.