Why does only one person "teach" during a church meeting?

essentially lots of people talking == church split

There is no real Biblical basis for only one person teaching during a church service. Such techniques were developed in response to problems that existed when a plurality of people are perceived as leading the church. (Especially if they are old people or snobby in such a way it attracts the weak) Its the Ephesians and Corinth problem that Timothy and Paul historically faced. If you are unfamiliar with this problem I will provide a longer explanation.

The Church as with any other entity has gone through the process of evolution.

In its infancy the church was established by word of mouth, by that I mean that in the absence of any written directives, churches were formed on the basis of the teachings of Jesus and, the story of his sacrifice being passed around through word of mouth.

In its infancy Paul and Silas along with others spread the word throughout the civilized world at the time. Later the Gospel was preserved in writing and Paul's guidance was slowly disseminated through his letters.

In the Church there was no collection of Scripture, which would give stability to church teachings, and as a result many varying ideas were slowly introduced into the concepts of the way in which the salvation of man was initiated by Jesus. This resulted in many, varying and quite frankly, dangerous misconceptions.

Some of these were the incorporation of the concept of reincarnation, which has no basis in Jesus teachings; Gnosticism , and an array of misinterpretations.

The Church went through a agonizing process of sanitizing the basic concepts of Christianity, and in the council meetings at Nicene established a common canon for use in the Church, which I personally feel was God's will.

Under the rule of Constantine in Rome this was required for Christianity to be adopted as the state religin of the Roman Empire.

At that time the Roman Empire ruled most of the civilized world, and all of the Roman Empire was required to teach only this and no other ideology. To this end the clergy was the path for dissemination of the standard teaching, and thus began the practice of one teacher in the Church, this responsibility was placed on the Cardinals who attended the conferences, and they in turn taught the priests. At that time there was no division of the Church.

Even as the church grew and went through its various schisms the concept of one trained teacher was incorporated into the break away factions. These break away factions, over the years have produced their separate canons, and have established their own institutes for the qualification of their ministers.

While the canons of various denominations differ widely, the concept of salvation through the death and resurrection of Jesus has held firm, and is the basis of all Christian denominations.

I believe what you cited is the intended example and would love to experience some of the things mentioned by @Steve. However, there are some examples of a single speaker, including just about every time Jesus spoke publicly after a certain point:

And from then on no one dared ask him any more questions. (Mark 12:34b)

The primary example of a single long monologue seems to be Acts 20:7-12, where Paul speaks all night long (and possibly all day and then all night long).

Fundamentalist Humor: Of course when I'm talking to someone who may be preaching soon, I tell them that they're quite welcome to talk until midnight just as soon as they demonstrate that like Paul they can raise someone from the dead.

In my opinion the single person monologue was primarily used for special occasions when an important speaker was in town for a limited time.

I attended meetings of the Vineyard for a period of a year and the members there exercised the gifts of the Spirit, with multiple people, one at a time, giving revelation and prophesying, tongues and interpretations. Then they bunched together and prayed for healings for the afflicted.

I attended a Brethren-inspired gathering for about 8 years. There, they encouraged multiple brothers to get up and speak on the same Sunday morning. The one with little experience spoke for 10 minutes; those with more ability, 15-20 minutes; those with gift, 30 minutes. (So when I say that I have preached for 8 years, people think I'm a pastor!) In this same church, the singing was not led by a choir or worship leader, but members of the body spoke out a hymn number between individual bouts of verbal worship, and we sang that.

Because these settings are outside of any traditional settings I've witnessed in mainline denominations, I can only conclude that historical tradition has dictated the idea of a single speaker. Also, it makes sense for the pastor to do all the speaking because he is paid to do so and (generally) has the ability to communicate the things of God well. In a regular church service where many are present, it certainly makes for a more controlled environment when the message isn't interrupted with questions or comments (I've attended one of those, too!)