The New Testament tells us four reasons why there are no apostles after the first century.
1. Christ's original apostles were only twelve in number (Matthew 10:1-4). Judas Iscariot lost his place because of his wickedness, but God replaced him with Matthias (Acts 1:15-26). At the end of time, the Lord symbolically says there are still only twelve (Revelation 21:14).
A short time after Pentecost, God chose Paul to be an apostle (Acts 9:1-20). Paul didn't join the Twelve; though he was equal in authority to them (Galatians 1:1). It also appears Barnabas was an apostle, prior to Paul, in light of Barnabas being called by that title in Acts 14:4.
2. The purpose of the apostles was to found the Christian faith, not merely to spread it. The apostles were not just church-planters. The apostles, along with the New Testament prophets, were the foundation of the Christian Church (Ephesians 2:19-20). This is why the New Jerusalem in Revelation 21 pictures them as foundation-stones. This function made their role unique.
3. Paul said he was the last of the apostles, because he was the last to see the resurrected Christ (1 Corinthians 15:8). This implies that Paul saw Christ's actual physical body, while on the road to Damascus. Paul had been in ministry for several years when he wrote 1st Corinthians, and yet (according to him) no one had seen the risen Christ after he did. Paul was the last.
4. The position of "apostle" does not pass down from anyone to anyone. The Roman Catholic doctrine of apostolic succession has no Biblical basis behind it. The "rock" upon which Christ builds His church is Peter's confession of faith, not Peter himself. Christ said, "upon this rock", not "upon you." There is no rule that says that "apostle" or "prophet" are continuous positions, when there is positive Biblical evidence that they (unlike evangelist, pastor, and teacher) had time-limiting attributes.
So, the original twelve apostles were a closed set. No more than twelve, and no more than those twelve. God then ordained two more (Paul and Barnabas) during the founding of the Church, in order to found the Church. But the Church has been thoroughly founded; it doesn't need any more founding, and it isn't being continuously founded a little bit every day.
The New Testament ended with the book of Revelation, with a warning not to add anything to it. God may work signs through various preachers over the years, but that doesn't make those preachers apostles; it just means God gave them a spiritual gift. The word apostolos is sometimes used in the New Testament in the generic sense of "messenger", like when someone brought a message or gift to Paul from a church. But we speak here of "apostle" in the specialized sense with which we are familiar.
There are no apostles, because the apostles finished their work. They established the Gospel by their eye-witness, and wrote the New Testament as our rule of faith. Their work is then carried on through the years by evangelists, pastors, and teachers.