“burned up.” The Greek is katakaiō (#2618 κατακαίω), and means to burn up, or to consume. It cannot be overstated that the weeds “burn up,” they do not burn forever. Similarly, the people who are unsaved will be burned up in Gehenna, they will not burn forever.
“the end of the age.” The Jews taught that we are in the present evil age, but there was a wonderful Messianic Age coming in the future. Although some versions read “world” instead of “age,” that is misleading. The world will not come to an end, but this evil age will. See commentary on Galatians 1:4.