"For Christ did not send me to baptize, but to preach the gospel - not with words of human wisdom, lest the cross of Christ be emptied of its power." NIV translation
Paul reminds us that it is all about sharing the fact that Jesus died as payment for our sins and rose again. He reminds us that, if he could have done it on his own, then there would have been no need for Jesus to die on that cross. Because Jesus rose again, death has been conquered and God's power displayed for all the world to see. It is that same power that gave Paul and gives us the ability to preach the gospel.
"For the message of the cross is foolishness to those who are perishing, but to us who are being saved it is the power of God. For it is written: 'I will destroy the wisdom of the wise; the intelligence of the intelligent I will frustrate.'" NIV translation
Without the Holy Spirit, it is not possible to understand the things of God and so the fact that the Son of God would come and die on a cross is foolishness to those that do not have the Spirit. Those that rely on their own wisdom are the hardest to reach with the gospel (just try to tell your college professor about Christ). This is nothing new as Paul quotes from Isaiah 29:14 and we see that God has already told us of this.
"Where is the wise man? Where is the scholar? Where is the philosopher of this age? Has not God made foolish the wisdom of the world?" NIV translation
There have been many so-called great thinkers throughout the history of man. They have come up with many ideas (religions) to have a right relationship with God. Every one, except faith in Jesus, relies on the effort of man to get to God. To them, it is foolishness that God would sacrifice His only Son to make a way. It's too easy and so they come up with another ritual or list of things to do but, in the end, those ideas lead to death (continued separation from God).
"For since in the wisdom of God the world through its wisdom did not know him, God was pleased through the foolishness of what was preached to save those who believe." NIV translation
These same scholars were the ones that did not recognize Jesus as the Messiah that they were looking for. He was not the military leader or earthly king that they expected and so they did not accept Him. Even though they did not understand it, God went ahead and sent His Son to the cross for us and them. Even today, people can get hung up on their own wisdom and not believe but, to those who do, this "foolishness" becomes the ultimate gift of grace.