I'm truly sorry Man's dominion Has broken Nature's social union,
There's no real break in this stanza—it's all one long sentence, and a real mouthful. But for the sake of convenience, we'll pause for breath after the first two lines of the stanza.
The speaker now apologizes to the mouse, but not just for accidentally plowing over its nest.
He apologizes on behalf of all humankind.
He says that he's sorry that the rule of man in the world has broken the "social union" of nature.
What does that mean, though? According to the speaker, it sounds like before mankind was more-or-less in charge of things, the natural world lived in social harmony.
"Nature's social union" stands out from the rest of the stanza because it's written in standard English, instead of in the Scots dialect of the rest of the poem.
(Yes, Burns could speak in regular English, too—he just chose to write most of his poetry in Scots.)
"Social union" sounds like a reference to the eighteenth-century Enlightenment philosophy of the social contract, held by guys like Thomas Hobbes, Jean-Jacques Rousseau, and John Locke. So the speaker is thinking of himself not just as an individual, but as a representative for all of mankind, and he's apologizing to the mouse, who is a representative of the whole natural world. Whoa. No pressure, mouse.
An' justifies that ill opinion, Which makes thee startle, At me, thy poor, earth-born companion, An' fellow-mortal!
The speaker continues his apology to the mouse, as a representative of all of mankind: he says that because mankind broke the natural harmony in the world ("nature's social union"), the mouse is actually justified in his bad opinion of humans in general and the speaker in particular.
He says that the mouse's distrust of all humans is what makes him startle back in fright at the sight of the speaker.
The mouse is afraid of the speaker even though the speaker isn't a threat—like the mouse, he's just a poor inhabitant of the earth. Like the mouse, he's a "fellow-mortal," or something that's going to die one day.
Wow, the speaker is claiming that he has a lot in common with the mouse.
We should take a moment to point out that in the Scots dialect (and in more old-fashioned forms of standard English), the words "thee" and "thou" and "thy" are used in place of "you" in more informal settings. This might come as a surprise—we tend to think of old-fashioned words as being more formal, but "thee" and "thou" were actually used only with members of the family and people that you knew very intimately. So by saying "thee" to the mouse, the speaker is being very friendly and familiar.