When Richmond personifies the sun as "weary" he is indicating that it is almost time for the sun to set. The sun is "weary," or tired, because it has been up all day. This is Shakespeare's way of telling his audience what time of day it is—evening. Shakespeare's plays were originally written to be performed in the Globe Theatre in London. This was (and still is) an open air theatre, and there was of course no electricity, so Shakespeare had to use speech and props (like candles) to indicate the time of the day to his audience.

When Richmond says that the sun has "made a golden set," he is referring to the setting of the sun. The sun is setting over the horizon and the sky appears to be a golden color.

Richmond then refers to "the bright track of (the sun's) fiery car." This is an allusion to the Greek myth of Helios. According to the myth, Helios was the sun god who rode in a chariot, or car, and pulled the sun behind him across the sky. The fact that the sun, pulled by Helios, seems to leave a "bright track" behind it is interpreted by Richmond as a good omen for the upcoming battle. He takes this "bright track" as an omen, or token, of a "goodly day to-morrow."

During this speech Richmond is trying to enthuse and inspire the men who will be fighting alongside him. Richmond thus interprets the brightness of the sun that day as a good omen in part to inspire these men. He needs them to go into battle with confidence that they will be victorious.