Brutus is saying:
What are you trying to tell me? If it is something that will advance the general good, then it makes no difference to me whether it brings me honor, death, or neither; I will do it regardless. If it is true that I love honor more than I fear death, may I have the blessings of the gods for that endeavor!
A line-by-line explanation of the lines in bold:
If it be aught toward the general good
If what you are trying to urge me to do benefits humankind; if it is for the greater good of humanity. Aught means anything, and its ambiguous placement here make the sentence mean both “If anything you are suggesting will advance the greater good” and “If what you are suggesting will do anything for the common good.”
Set honor in one eye and death i’ th’ other
Then put honor and death equally in front of me, as though out of one eye I can see honor before me, and out of the other I can see death before me
And I will look on both indifferently;
And I will regard them both equally; I will not care about one more than the other; I will not let the fear of death keep me from acting honorably; I will not act dishonorably to escape death; I will act to benefit humankind irrespective of whether my action brings honor, death, both, or neither. This line is very tightly packed because of the double meaning of indifferently, which means both equally (not giving more weight to one over the other) and stoically or unemotionally (without having any personal stake in the matter). The word impartially is probably a good equivalent, since it captures both these senses.
For let the gods so speed me
May the gods set me up to succeed in this endeavor (the endeavor for the public good). Speed is used as a transitive verb in sense 2 in Merriam-Webster: “to further the success of” or “to cause or help to prosper.”
... as I love
The name of honor more than I fear death.
... to the extent that my love for honor outweighs my fear of death. So ... as means to the extent that. See this Grammarphobia article that says "'so … as' constructions indicate extent or degree." See also this question in English Language Learners and the accepted answer, which says "the so... as ... construction is used ... to introduce a measure of the extent to which the preceding assertion is true."
To further clarify the last two lines: In the earlier lines Brutus has said that if it's a matter of acting in the public interest, he will do that regardless of whether he gets honor, death, both, or neither as a result. In this last line he is emphasizing that his love of honor is indeed greater than his fear of death by swearing on that fact: "If it is true that my love of honor is greater than my fear of death, then I pray that the gods support me in my attempt to act for the greater good, regardless of the consequences to myself."
Edit. OP asks in a comment:
what’s still troubling me is the word “for” at the start of line 7. It makes it sound like, BECAUSE Brutus loves honour more than he fears death, he will look at both with indifference - which doesn’t make sense to me!
Brutus is saying that even though he loves honor more than he fears death, when it's a question of the public welfare, he will act without regard for either. For here doesn't mean because; it is introducing a prayer or an oath being sworn, not a reason. See meaning 5 listed in the definition of for at dictionary.com:
(used to express a wish, as of something to be experienced or obtained):
O, for a cold drink!
A longer paraphrase of the last two lines, drawing on what Brutus has said before, would be something like this:
Since it is true that I love the name of honor more than I fear death, then to the extent that it is true, I pray that when it comes to public welfare, the gods grant me aid and success (speed me) in my endeavor to act without regard (indifferently) toward both death and honor.
Two points are noteworthy:
- Brutus is speaking paradoxically here, admitting that he loves honor more than he fears death, yet asking the gods to let him treat them both with equal disregard. The prayer swears upon something while simultaneously asking for that thing to be destroyed. This makes the passage difficult to understand. The crux is that Brutus is saying: "I love honor. I fear death. But if I have to undertake an action for the good of Rome, I will do it with equal disregard for both." And then he is praying to the gods to let him be successful at acting in this way.
- Brutus specifically says the name of honor, not just honor. His point is that truly acting for the greater good is honorable whether or not the act gets us renown. Doing something simply to gain honor could be a dishonorable act—another paradox. So he is saying: "Yes, I like renown and I fear death. But I hope the gods will give me the ability to put both of these aside if and when I have to act for the public welfare."
Finally, a really simple way of looking at the last two lines could be merely that because Brutus likes honor and fears death, he is praying to the gods to give him the strength to act objectively, so that what motivates him is the good of Rome rather than his own desire or fear.
