Do your best to present yourself to God as one approved, a worker who has no need to be ashamed, rightly handling the word of truth. (2 Timothy 2:15)
Exegesis literally means “to lead out of.” Exegesis is the exposition or explanation of a text based on a careful, objective analysis. It's the process of discovering the original and intended meaning of a passage of scripture. The opposite approach to Scripture is eisegesis, which is the interpretation of a passage based on a subjective, non-analytical reading.
What does the passage say? What does the passage mean? How does the passage relate to the rest of the Bible? How should this passage affect my life? You need to understand the context - the genre, the surrounding story, the character, the current events and surrounding society, the author, the intended audience, the specific lesson being conveyed at the time, etc. You also need to understand the underlying original language and the accompanying culture and perspectives. You also need to apply it.
(https://seminary.grace.edu/what-is-exegetical-study-and-how-can-you-use-it/, https://www.gotquestions.org/exegesis-eisegesis.html)
This method removes apparent errors in the Bible, like the genealogies of Jesus (Matthew 1 & Luke 3), the accounts of Paul's conversion (Acts 9, 22, 26), errors of fact (rabbit chewing cud in Leviticus 11:6), and the sun standing still (Joshua 10:12-14). Logical and scientific difficulties remain, but are they errors, contradictions, faults of a copyist or translator, or a problem of the cultural, historical, or rhetorical gap between writer and reader? (The Origin of the Bible: Newly Updated by F. F. Bruce, J. I. Packer, Philip W. Comfort, and Carl F. H. Henry, 2020. Page 41.)
Exercise
The interpreter reads 2 Chronicles 27:1-2 and sees that King Jotham was a good king, just like his father Uzziah had been, except for one thing: he didn’t go to the temple! First, the interpreter reads the passage and, to fully understand the context, he reads the histories of both Uzziah and Jotham (2 Chronicles 26-27; 2 Kings 15:1-6, 32-38). In his observation, he discovers that King Uzziah was a good king who nevertheless disobeyed the Lord when he went to the temple and offered incense on the altar—something only a priest had the right to do (2 Chronicles 26:16-20). Uzziah’s pride and his contamination of the temple resulted in his having “leprosy until the day he died” (2 Chronicles 26:21).
Needing to know why Uzziah spent the rest of his life in isolation, the interpreter studies Leviticus 13:46 and does some research on leprosy. Then he compares the use of illness as a punishment in other passages, such as 2 Kings 5:27; 2 Chronicles 16:12; and 21:12-15.
By this time, the exegete understands something important: when the passage says Jotham “did not enter the temple of the LORD,” it means he did not repeat his father’s mistake. Uzziah had proudly usurped the priest’s office; Jotham was more obedient.
The resulting sermon might deal with the Lord’s discipline of His children, with the blessing of total obedience, or with our need to learn from the mistakes of the past rather than repeat them.