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An excellent question, and one that I've pondered myself in the past.
A pastor I knew once actually gave a sermon on the issue. His explanation paraphrased is that Uzzah thought that the ark touching him would not sully it, while it touching the ground would.
Uzzah's actions, though seemingly innocuous enough, stemmed from a fundamental misunderstanding of his and the Lord's identity.
If you look at the context surrounding the verse carefully you'll see numerous infractions already committed by Uzzah and his kin. For instance, the Ark was being pulled along in a wagon despite the law saying that only levites were to carry it on wooden rods - Deuteronomy 31:25 & Exodus 25:14-15. Uzzah had become complacent and at the worst possible time - when the ark was being re-introduced to Israelite society. For God to have allowed these insults to pass would have set a precedent among Israel that the throne of God does not need to be treated with reverence and honor.
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The implication of Numbers 4:15 is that God forbids non-priests from touching holy things like the Ark of the Covenant. Apparently, God killed Uzzah because he, not being a priest, touched a holy thing contrary to God's commandment.
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We mustn’t read more into the text than the Bible allows. The Bible reveals Uzzah’s physical death the Bible does not reveal his eternal outcome. He may have passed into the presence of the Lord.
The Arc of the Covenant was the dwelling place of God; Uzzah’s error was attempting to save God by steadying the arc. God is the Savior of mankind not the other way around.
Could you imagine Uzzah walking around and everyone whispering “there is the man that saved God”.