“No one can come to Me unless the Father who sent Me draws him” John 6:44
What did Jesus mean by this statement? Most basically what he meant was just what He said. Unless a person undergoes a new birth they cannot either see or enter the Kingdom of God (John 3) and that new birth is not of blood nor of the will of the flesh nor of the will of man but of God (1 John 1:13). Fallen humanity, apart from the drawing power of God, does not seek after Christ nor can they or will they come to Him. Strict Armenians are stonewalled here by this unwavering statement of God's sovereign and necessary involvement in the salvation of men but how to tone down the Calvinists denial of human choice so clearly invited by Jesus in Matthew 11:28-29 among others? "Come to me, all who labor and are heavy laden, and I will give you rest. Take my yoke upon you, and learn from me, for I am gentle and lowly in heart, and you will find rest for your souls.
If I may I will summarize the remainder of your questions as, How does this drawing work while maintaining both God's sovereignty and human choice?
First I suggest that these are not mutually exclusive. God sovereignly created human choice in the Garden of Eden and protects this faculty within us even when it most often leads to our demise. I call heaven and earth to witness against you today, that I have set before you life and death, blessing and curse. Therefore choose life, that you and your offspring may live - Dt 30:19 In the next verse choosing life is equated with loving God and obeying Him. It is His desire for a relationship based upon love that has necessitated our possession of a faculty of choice.
Sin has ruined our prospects for making such a choice. It is not that the choice to seek after and love the Lord has been removed by God, else the invitations peppered throughout Scripture to "return unto Me" (Zech 1:3, Mal 3:17, Jer 15:19, John 7:37, etc.) would be a cruel joke. Rather our natural disposition "in Adam" is to reject God's authority to determine good and evil and to take that mantle upon ourselves and thereby "be like God". There in the garden, while Adam and Eve hid, God sought them out and began questioning them. He drew them out of hiding...although they possessed the ability to do so they did not come until he called. "So the Lord God called out to the man, "Where are you?"". Genesis 3:9 The earth then was cursed "for our sake", for our benefit that is, because having rejected God in Paradise we would certainly not seek after Him absent difficulty. God draws us through the difficulty of life in a cursed creation.
Such is the condition of every person, enslaved by sin (Romans 6:16) and fear of death (Hebrews 2:15) they are unable to please God (Romans 8:8) or to desire to do so "For the mind that is set on the flesh is hostile to God, for it does not submit to God's law; indeed, it cannot. Those who are in the flesh cannot please God. - Romans 8:7-8
We could come to Christ apart from the Father's drawing us but we don't want to because we understand that it requires submission. Here it approaches the level of cannot but the prohibition is not from God as the Calvinists suppose. The inability of the natural man is one of volition: He will not.
Here lies the explanation of the unbelieving Jews of Acts 13 and all others who hear and refuse to believe: "You search the Scriptures because you think that in them you have eternal life; and it is they that bear witness about me, yet you refuse to come to me that you may have life." - John 5:39-40 Natural man wants to obtain life through some measure of intellect or effort and refuses to humbly come and receive.
Jesus said that if/when He was lifted up He would draw all men to Himself (John 12:32) and it is then explicitly stated that he said this in reference to the method of his death. It is fair to say, then, that the means by which God the Father draws people to come to Jesus is by sending His Son to be lifted up on the cross. Remembering that Jesus told Nicodemus he must be lifted up just as (in the same way that) Moses lifted up the bronze serpent we see both sovereign drawing, divine provision, and human choice in operation. In Numbers 21 we are told that those who looked upon the bronze snake lived: apparently some did not look.
But how is the Son of God on the cross "just as" a bronze snake on a pole. The venemous snakes were sent as judgement and bitten people were dying. Bronze speaks of judgement (both divine and self) and the bronze image of what was killing them was lifted up so that when they looked at it (with faith) recognizing their culpability and God's justice they could receive His mercy. Jesus is the Word of God made into flesh (John 1:14) and the Word of God on the cross is an image of what is killing us. Adam put God's Word to death in himself as the prime guiding principle for life and it is that death that we carry about in our bodies. The Word of God was lifted up on the cross so that we may recognize our participation in that which is killing us, that we may repent of it, and that we may come to Him in order to have life.
In short, God draws us to Christ by sacrificing Him. Jesus draws us to Himself by that act of obedience. The Spirit has come to persuade: "And when he comes, he will convict the world concerning sin and righteousness and judgment: concerning sin, because they do not believe in me; concerning righteousness, because I go to the Father, and you will see me no longer; concerning judgment, because the ruler of this world is judged. - John 16:8-11
Faith comes by hearing and hearing by the Word of God. The gospel is the power of God for the salvation of those who will believe. The proclamation of the gospel exposes the hearer to the drawing power of God in the crucified Son and those who choose to believe are exposed to the saving power of God in His resurrected life.