The wild dates assigned to the Didache are most likely a result of the the several developments it seems to have gone under. Some scholars have even attempted to remove the "additions" to the original Didache in order to uncover the original text1. It is difficult to determine just how late the last development occurred, so for this reason, I will be assuming that you are asking when the Didache in its original form was first composed.
The importance of the Didache stems from the fact that some of the early Christian churches considered it to be canonical. Even some modern-day Oriental Orthodox Churches consider a derrivative of the Didache, the Didascalia, to be part of a broad canon2. Though generally the Didache and her derivatives have not been considered canonical by Christendom in general. This is likely due to the very practical nature of the document.
This answer will quickly summarize the various arguments put forth and allow you to follow up on each of these points for further study. It will not offer arguments for a late date, as this is beyond the scope of your question.
Arguments for a 1st Century Dating
- The primitive church presented by the Didache is highly indicative of an early composition date. The Didache shows little to no knowledge of the more formal structures referenced by those like Clement of Rome and Ignatius of Antioch (1 Clement dating ~70-140 CE and Letter to the Smyrnaeans ~107 CE). Some believe the Didache to have been composed in Antioch, which then begs the question why it would not mention the formal church structures Ignatius (who lives in Antioch) mentions.
- The Didache seems to lack Pauline influence which indicates its writers where not familiar with Paul's theological ideas expressed in his letters dating ~50 CE onward. This leads to the conclusion that the Didache was written before its writers had received the famous Pauline letters we know today.
- The manner in which the sayings of Jesus are presented suggest they were received by oral tradition. This has led some to believe the Didache lacked all four of the written gospel accounts. For example, the golden rule is recited in the negative ("do not do to another what you would not want done to you" Didache 1.23) unlike the gospels which recite it in the positive. This suggests that the writers were relying on oral tradition rather than written tradition.
- The Didache begins by arguing that there are "two ways." This comports with the early Christian name "those who follow the way" (Acts 9:2; 19:9, 23; 22:4; 24:14, 22) and from Jesus' teaching of the two gates (Matthew 7:13-14). Jesus is also called the way (John 14:6). This is highly suggestive that the original writers of the Didache still considered Christianity to be a sect or "way" within the broader religion of Judaism4.
- Dr. Alan Garrow argues the Didache is a possible solution to the synoptic problem5 which would date it before the Synoptic Gospels.
- Several parts of the Didache demonstrate it is targeted towards gentiles, not Jews. This has led some to consider it an early catechism used by Jewish-Christians to first evangelize the gentiles. Essentially, it is a vague suggestion that the Didache is one of the first attempts by the original Jewish-Christians to evangelize the gentiles.
- The Didache shows little sign of contemporary persecution in the writters' Christian communities. This suggests it was written before the harsh persecution characterizing the 60's CE.6
- There is no "polemic against heterodox or Gnostic tendencies within the church."6 This suggests the Didache was composed before heterodox and Gnostic tendencies appeared within the church.
- "The Didache does not fit clearly into any period of liturgy or ministry for which we have documentation."6 The reason may be because it was written before any liturgy we know of had formed.
- In chapter 6 there is an appeal to abstain from food offered to idols. This comports with James' admonition in Acts 15 at the council of Jerusalem (~50 CE) and is unlike teachings after the council on the subject (1 Corinthians 8:4-6 ~53 CE). This has also led some to identify the Didache with the Apostolic Decree written by James the Just.
- Despite the apocalyptic ending, the Didache makes no connection to the temple destruction. However, it is hard to tell if this indicates much.
- The longer name of the Didache is "The Teaching of the Twelve Apostles." The even fuller title is "The Teaching of the Lord to the Gentiles by the Twelve Apostles." If this title were correct, it would have been written by the apostles themselves.
- Some have argued that the Didache is a "matriarch" of Apocalypse of John (or Revelation), which if true, would naturally date the Didache before John wrote his apocalypse. See Dr. Alan Garrow's discussion7.
I have presented one side of the argument because this is what your question asked for. However, note that the Didache is one of the most wildly dated Christian documents, and it is likely that it will remain so. It is a document that many Christians esteem but rarely consider canonical.