The disaster occurring at this time is spoken of as if the day itself were burning. God's judgment is often spoken of as if it were a fire. (See: [[rc://en/ta/man/translate/figs-metaphor]])
These people are spoken of as if they will become dried-up plants fit only for being burned. It is common for the Bible to speak of people as if they were plants or trees. Alternate translation: "all the arrogant and all the evildoers will burn up like dry plants" (See: [[rc://en/ta/man/translate/figs-metaphor]])
Here "the day" stands for the events that will occur on that day. Alternate translation: "On that day I will burn them up" (See: [[rc://en/ta/man/translate/figs-metonymy]])
This expression continues to speak of people as if they were plants or trees. So being deprived of all roots and branches stands for being completely killed off. Alternate translation: "nothing will be left" (See: [[rc://en/ta/man/translate/figs-merism]] and [[rc://en/ta/man/translate/figs-metaphor]])