Paul is giving his own reasoning for what he allows and does not allow personally. It is plain from the reading of this passage that this is the policy which he follows and instructs others, but note that he is not presenting this as settled church teaching. Please also note that while in other passages, Paul makes it clear that he is speaking on behalf of the Lord, but in this case he does not. In this case, it is entirely possible that Paul is reasoning from his own training as a Pharisee and from within his own historical and cultural context as an Israelite in the ancient Near East, as references by another answer that has been given.
Remember that in the early church, there are many matters that are not fully settled, including crucial issues such as the Trinity, the nature of Christ's humanity and divinity, the role and inclusion of Gentiles in the church, and whether Jewish sacrificial law should continue to be followed. Since we trust that the Holy Spirit leads the church "into all truth" as Jesus has told us (John 16), we can rely on the decisions reached by Apostles and the church fathers in the Ecumenical councils held first in Jerusalem and then subsequently elsewhere, and we can see that the church's understanding of these key issues is clarified and confirmed over time. Please note here that the church has always taught that while the message given to us in Scripture does not change, our understanding of it develops over time as we seek deeper understanding of what God has revealed.
Please also note that there are many instances in Scripture where people say or do things that don't necessarily reflect the fullness of truth. Sometimes, we are given examples of bad behavior, and sometimes, people are walking by the light they have at the time. Just because Scripture presents a statement or a story to us does not mean that we are being taught that we should do this as well. This is where it can become helpful to us to have a trustworthy church authority that helps us understand what Scripture teaches (as distinct from what it merely presents to us as examples), and what God is asking us to do.
As a parallel example that may be helpful, I would suggest reading the progression of Peter in the book of Acts on how God sees the Gentiles (especially Acts 10, Acts 15, Galatians 2). At first, he is in opposition to God's view of the Gentiles, and is opposed on this point by Paul. However, when he receives the vision ("call nothing unclean which God has made", a reference to the Gentiles, not only foods not allowed to Jews), and the mission to go baptize Cornelius, where he sees the Holy Spirit come upon the Gentiles, he allows God to convince him that he is wrong on the subject, and later preaches a very different message at the council of Jerusalem. Finally, with Paul, Peter, and James in agreement, the council affirms what God has revealed to them and writes a letter to the non-Jewish believers.
This understanding of how church teaching develops under the superintention of the Holy Spirit as expressed through the legitimate authority of the Apostles and their successors is a big part of the reason why a number of ancient church traditions, including mine, do have women as teachers, and yet do not violate what the Scriptures assert.
Please note that I am not saying that Paul is "wrong" here. I am simply suggesting that he, as well as the entire church, are still working out what the role of women, Gentiles, etc. looks like in Christianity, and how to approach this from a practical and pastoral perspective that also does not lead to scandal or confusion among the Jews and Gentiles in the church.