What is the purpose and significance of allegory in The Divine Comedy and how does it convey the truth like Dante claims?

Great question! It is important to define the term "allegory" before answering your query.
An allegory is a story that can be read on two levels: the literal and the symbolic. The literal level of an allegory is the setting, plot, and characters of the tale; I often explain to my students that even young children can understand the literal plane of most allegories. The symbolic level is the secondary interpretation of the story, where the plot, characters, and setting take on a figurative meaning that communicates a message, normally political or philosophical in nature.
While it is possible to read Dante’s The Divine Comedy as a straightforward story about a journey through hell, purgatory, and heaven, many scholars argue that Dante Alighieri’s three-part epic is an example of religious allegory. This is because it communicates a stark religious message about the essential state of mankind and outlines a spiritual solution to humanity’s woes. Perhaps breaking the trilogy into its three individual parts will make it easier to show examples of how The Divine Comedy is clearly allegorical.
The poem begins with Dante finding himself, halfway through his life, lost in a dark forest. He encounters Virgil, who promises to guide him to safety through the realms of the afterlife, starting with a tour of hell. In hell, Dante encounters kings and killers, poets and paupers all suffering in inventive ways for their sins. The allegorical message of Dante’s journey through hell is clear: all humans are sinful and, without God’s aid, will end up here.
In “Purgatorio,” Dante travels to the realm of Purgatory with Virgil and eventually switches guides to Beatrice, as his Roman-poet friend was not a Christian and cannot proceed any higher into the realms of the afterlife. In “Purgatorio,” Dante highlights how a life of penance can bring a human soul closer to God. This is particularly well-exemplified in the three steps at the entrance to Purgatory, where they represent the need to recognize sin in order to combat it and repent from it.
Finally, in “Paradiso,” Dante explores Heaven with Beatrice and ascends through nine levels of virtue adopted by souls who inhabit the divine kingdom. These include Justice, Temperance, Love, and Faith. Through this portion of The Divine Comedy, Dante explores the Christian values and beliefs required to enter Paradise and, by extension, promotes these ideals through his allegory.
As an allegory, “The Divine Comedy” conveys Dante’s ultimate religious truth: that all humans are sinful and destined for hellish eternity unless they trust in and obey God. I hope this helps!