“He offered the tribute.” More literally, “He brought near the tribute.” The text is using sacrificial language.
“a very fat man.” Eglon is the only man who is named and specifically called “fat” in the Bible (although the High Priest Eli is called “heavy,” 1 Sam. 4:18). And Ehud is the only man in the Bible who is named and called “left-handed.” So this is a conflict between two unique men: lefty versus fatty. Looked at another way, being left-handed was a terrible cultural disadvantage and considered a curse, while the fact that Eglon was fat showed that he had the best of life: a sedentary lifestyle and loads of food. So this record subtly shows that being disadvantaged but with God is always better than being worldly advantaged but without God.