[...] and there the truth divine be heard through me, unworthy as I am (15-16)
Petrarch speaks not only of the duty of princes, but also of the obligations of the poet. It's really his job to tell it like it is—even if the lords don't like what they hear. It's also a way for Petrarch to wiggle out of the nobility's wrath when they realize he's dissing them.
I speak to tell the truth, not out of hate or scorn for anybody (63-64)
And again, Petrarch is going with the truth thing. He's trying to show that he doesn't have an ulterior motive—like political or monetary gain—in the writing of this poem. He wants the nobility to know that he has to speak as he feels, because things are bad.
From dawn to tierce examine yourselves, and you will see that one who thinks himself so cheap cannot hold others dear (71-73)
For one thing, Petrarch says to the Italian princes, you've got a duty to meditate on your motives for wreaking such havoc. It's also their burden to consider the lives of their subjects and neighbors before they make the decision to bring foreign hostile forces into play.
O noble Latin blood, release yourself from such a harmful burden, and do not idolize a name that is so empty (74-77)
Petrarch is really pushing the nobility to understand their civic duty to the people of their lands. For one thing, they've got the noble lineage (i.e., ancient Rome) that should allow them to make good decisions on behalf of the people. For another, they should be sensible enough not to make war over something that isn't for the benefit of society at large.
In God's name may your mind for once be moved by this, and look with pity upon the tears of all your grieving people (87-89)
A good prince, Petrarch implies, would listen to the pleas of his people. If he can't act out of love for his people, he needs to go back to nobility school.
[...] and all the time you spend in giving others pain, to some more worthy act of hand of intellect to some beautiful praise, to worthy dedication be converted (106-111)
Here's a thought: if Fortune favors you to be born part of the ruling class, you should probably act like a noble person. Petrarch wants to slap the nobility upside the head for their (self) destructive behaviors and urges them to turn their minds to something more productive and optimistic.
But you must try your fortune among the valiant few who love the good; tell them: "Who will protect me? I go my way beseeching: Peace, peace, peace." (119-122)
Here's one way to get a medieval lord to think kindly of your poem: dress it up like a damsel in distress. Petrarch does just that in his congedo when he tells the poem that it's going to have to work hard to convince the nobility to grow up and step up.