Paul's point is that the Corinthians, by pursuing human wisdom, are acting just like those without the Spirit and are missing the very wisdom of God.
Paul's preaching is described as "not according to excellence of word or wisdom", i.e., not excellent according to popular understanding of wisdom in both form and content. Paul indeed speak wisdom, not esoteric knowledge of deeper truths about God but simply his plan for salvation.
Paul's singular focus and passion is on the gospel of Christ.
Paul's "weakness" probably refers to some observable physical condition. The apostle regularly glories in his weaknesses as a sure evidence that the power was of God and not of himself.
"Fear" and "trembling" occur together in the Old Testament (Ex 15:16, Is 19:16, from the Septuagint, the Greek Old Testament which Paul used for quotes), usually to express the dread that people are to sense in the presence of God and his activity. This combination is unique to Paul in the NT (2Co 7:15, Php 2:12, Eph 6:5). Here Paul seems overwhelmed by the task of evangelizing in this great city.
Even though he was weak and his preaching lacked "rhetoric" and "wisdom", yet it came from the Spirit and was thus powerful; their very coming to faith demonstrated the power.
The main lesson ("so that"): do not rely on human wisdom.
"rulers of this age" = leaders of this age in the broader sense (v.8), not demonic powers; "coming to nothing" = in the process of being abolished
God's wisdom (the gospel of the cross) was a mystery ("secret wisdom") — something formerly hidden from all human eyes but now revealed through Christ to his people by the Spirit. Through the cross, God's people are predestined by God to share in his own glory.
"deep things of God": "depth of the riches of the wisdom and knowledge of God" (Ro 11:33)
Principle of "like is known only by like": only God can know God. Just as the only person who knows what goes on inside one's own mind is oneself, so only God knows the things of God.
We explain spiritual things by means of, or with, the spiritual words taught us by the Spirit.
The man without the Spirit ("the natural man"): (a) do not accept the things that come from the Spirit of God, (b) think that the things of the Spirit are foolishness to them, (c) cannot understand the very things that the one who has received the Spirit can.
For Paul, "to be spiritual" and "to discern spiritually" simply means to have the Spirit who then enables discernment (judgment) of true wisdom.
The person who has the Spirit can judge or discern God's ways. Not necessarily all things, of course, but all things that pertain to the work of salvation. Yet, the one without the Spirit cannot make judgements on the person with the Spirit.
The quote is from Is 40:13 meaning "Who is the person who wants to match wits with God?" making it clear that human beings are unable to comprehend the mind of God. But for those who have the Spirit or the "mind of Christ", they have the ability to understand.
In the Greek Old Testament (Septuagint or abbreviated LXX) that Paul cites, the word "mind" translates the Hebrew rûah, which ordinarily means "spirit".