What happens in Canto XV of Dante's Inferno?

Canto XV
Dante and Virgil are walking along the cooler edge of the burning sands that ring Circle Seven as they descend further into the center of Circle Seven. The travelers are protected by a fine mist that rises from the Phlegethon. The mist serves as a shield, extinguishing the flakes of fire that continually rain down in this circle, punishing those guilty of the crime of blasphemy and other acts of violence against God. The protective shield reminds Dante of the dams built around Italian cities for protection against seasonal flooding. This protective mist, strangely, is even more formidable than those heavy walls:

“...Even as the Flemings, 'twixt Cadsand and Bruges,
Fearing the flood that tow'rds them hurls itself,
Their bulwarks build to put the sea to flight;
And as the Paduans along the Brenta,
To guard their villas and their villages,
Or ever Chiarentana feel the heat;
In such similitude had those been made,
Albeit not so lofty nor so thick,
Whoever he might be, the master made them.”

While marveling about the mist, Dante notices a group of shades walking in the same direction in which he and Virgil travel. These sinners are the sodomites, those who have had sexual relations with other men.
As the condemned souls get closer to the poets, one of their number recognizes Dante, grabs onto Dante’s cloak and cries out. Dante struggles to recognize the man’s badly burned visage. Peering closer, Dante asks if the man is Brunetto Latini:

“By some one I was recognised, who seized
My garment's hem, and cried out, "What a marvel!"
And I, when he stretched forth his arm to me,
On his baked aspect fastened so mine eyes,
That the scorched countenance prevented not
His recognition by my intellect;
And bowing down my face unto his own,
I made reply, "Are you here, Ser Brunetto?"

The shade confirms the identification. Dante wants his former mentor to stay and speak, but the sinner explains that any soul who pauses or stops is punished in the same spot for one hundred years:

"O son," he said, "whoever of this herd
A moment stops, lies then a hundred years,
Nor fans himself when smiteth him the fire.
Therefore go on; I at thy skirts will come,
And afterward will I rejoin my band,
Which goes lamenting its eternal doom."

Dante agrees to walk along near Latini, although soon the path they traverse splits. One side goes along the lower edge of the river and is protected by the mist; the other is on a higher plane and it offers no relief from the falling fire. Dante is forced to take the lower and Latini the more elevated way.
Despite their separation, the two are able to exchange words. Latini wants to know how a living man is able to visit Hell; he also asks Dante with whom he travels:

"What fortune or what fate
Before the last day leadeth thee down here?
And who is this that showeth thee the way?"
Dante explains his wandering in the valley of darkness and Virgil’s part in assisting him:
"I lost me in a valley,
Or ever yet my age had been completed.
But yestermorn I turned my back upon it;
This one appeared to me, returning thither,
And homeward leadeth me along this road."

Latini tells Dante how fortunate he is to be so forewarned in such a vivid way about the eternal torments of Hell. He also laments his own death, in part because he is unable to assist his protege any longer:

"If thou thy star do follow,
Thou canst not fail thee of a glorious port,
If well I judged in the life beautiful.
And if I had not died so prematurely,
Seeing Heaven thus benignant unto thee,
I would have given thee comfort in the work.”

Thinking back to the travails of Earth, Latini tells Dante that his contemporaries, among them the Fiesoles (who had conquered Rome), failed to appreciate Dante’s genius. Furthermore, Latini, the former author and poet, and promoter of elegance in rhetoric, blames the Fiesoles for the decline of Florentine morals and values:

“But that ungrateful and malignant people,
Which of old time from Fesole descended,
And smacks still of the mountain and the granite,
Will make itself, for thy good deeds, thy foe;
And it is right; for among crabbed sorbs
It ill befits the sweet fig to bear fruit.
Old rumour in the world proclaims them blind;
A people avaricious, envious, proud;
Take heed that of their customs thou do cleanse thee.
Thy fortune so much honour doth reserve thee,
One party and the other shall be hungry
For thee; but far from goat shall be the grass.
Their litter let the beasts of Fesole
Make of themselves, nor let them touch the plant,
If any still upon their dunghill rise,
In which may yet revive the consecrated
Seed of those Romans, who remained there when
The nest of such great malice it became."

Dante’s love for his former teacher overflows. He praises Latini for all he taught him, primarily that the only true immortality for men comes through their work. Still attendant, Dante lets Latini know he is writing down whatever he says, in order that he might have Beatrice weigh in on his instruction. Finally, he assures the shade that he is prepared for Fortune’s arrows:

“For in my mind is fixed, and touches now
My heart the dear and good paternal image
Of you, when in the world from hour to hour
You taught me how a man becomes eternal;
And how much I am grateful, while I live
Behoves that in my language be discerned.
What you narrate of my career I write,
And keep it to be glossed with other text
By a Lady who can do it, if I reach her.
This much will I have manifest to you;
Provided that my conscience do not chide me,
For whatsoever Fortune I am ready.
Such handsel is not new unto mine ears;
Therefore let Fortune turn her wheel around
As it may please her, and the churl his mattock."

Virgil approves of Dante’s speech, nodding silently in agreement. As the three walk on, Dante asks Latini who else are among the sinners who suffer in this circle. But on this count, Latini is not very forthcoming. Although those who eternally shuffle along are legion, Latini names just three: Priscian, Francesco d’Accorso, and Bishop Andrea dei Mozzi.
Priscian could be one of two people. Scholars believe that the Priscian to whom Dante refers was either an early influential grammarian or he may be referring to a professor of that name who taught at law at Bologna. There may be more credence to the latter as Francesco d’Accorso was also a professor at Bologna. The final sodomite Latini identifies, Bishop Andrea dei Mozzi, had been transferred by Pope Bonfice VIII from Florence to Vicenza, where he died the following year.
Latini notices something that alarms him: smoke rising in the distance. He tells Dante he must go for these are people “with whom I may not be.” The shade fears the comers and wishes to depart, but first he asks if his own work, the Tesoro has lived on, thus giving his name, at least, literary immortality.
Before Dante can reply, however, Latini makes a hasty retreat; his speed reminds Dante of the famous races at Verona. The winner was awarded a “green mantle,” a scarf that wrapped over the shoulder and about the waist. Dante, watching the man depart, fantasizes Latini was won the famous prize:

“Then he turned round, and seemed to be of those
Who at Verona run for the Green Mantle
Across the plain; and seemed to be among them
The one who wins, and not the one who loses.”