Why did Jesus carry the cross and why did he say that everyone should take up their own cross? And what is the cross anyway?
I believe that we can get a definitive answer on this from the Apostle Paul. He spoke of the Cross as the summation of Christ and His Mission to redeem man.
He stated that the Message of the Cross is the Power and Wisdom of God to those who are saved (1 Cor 1:18, 23), and emphatically stated that he would not preach or know anything other than "Christ and Him Crucified" (1 Cor 2:2). He taught in many places that the very life we have as Christians stems from being planted together with Christ in His Death (Gal 2:20, Romans 6:6). If we compare what Paul says about the Wisdom of God from 1 Cor 1:23 and Eph 3:9-10, we see that he is wrapping up in the term "Cross" the entire, eternal mission of Christ. We can confirm that with Phil 2:8-9. So, when we are to take up His Cross, the Cross, our Cross, being that we are planted together with Him in the likeness of His Death, it means to take up His Redemptive Mission. We are to "seek first the Kingdom of God and His Righteousness" (Matt 6:33) (also flows from Calvary [Rom 3-4]). We are to live by faith in His Blood (Rom 3:25). We are to look to Calvary.
Since Grace flows through faith (Eph 2:8-9), and Satan knows that in order for him to remove us from the Grace of God, he must destroy our faith, he knows that he must eliminate our faith in the Atoning Work of Christ that reconciles us to God. Therefore, with believing properly comes a struggle of faith... a good fight of the faith. (1 Tim 6:12). This struggle (fight, literally "agony") to keep one's faith off of self (the world, the flesh, and the devil all fight against this faith) and unto Christ is what it means to "take up one's Cross". It means to carry the Cross in your faith, which will allow Grace (Eph 2:8-9), which will allow the Holy Spirit to conform us into the Image of Christ (Rom 8:29, 2 Cor 3:18), which will enable us to build the Kingdom of God in Resurrection Power (Acts 2:8; Romans 6:5).
Matthew 16:24 says "Then Jesus said to his disciples, "Whoever wants to be my disciple must deny themselves and take up their cross and follow me". Luke 9:23 says "Then he said to them all: "Whoever wants to be my disciple must deny themselves and take up their cross daily and follow me". Luke 14:27 says " And whoever does not carry their cross and follow me cannot be my disciple". And Matthew 11:30 says "For my yoke is easy and my burden is light".
In all these verses Christ is using Cross as an allegory to human suffering and sacrifice needed to carry out his responsibilities. Christ is alluding to the fact that as Christians and children of God our responsibility is to fulfill our daily responsibilities without any whining.
When Jesus said:
Luke 9:23 And he said to them all, If any man will come after me, let him deny himself, and take up his cross daily, and follow me.
it was some time prior to his actual crucifixion, and apparently he was making a reference to that crucifixion. His cross was the dying on the cross in payment; not for his sins since he had none; but for our sin. So if that was his cross to bear, what then is our cross?
Perhaps we find the answer to that in what Jesus charged his disciples with after he had born his cross. After he had risen from the Grave he gave his disciples only one command:
Matthew 28:18 through 20 KJV And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth. Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: Teaching them to observe all things whatsoever I have commanded you: and, lo, I am with you alway, even unto the end of the world. Amen.
Hope this helps.