Moving the Ark of God required strict adherence to the instructions originally given to Moses in the Torah [ביאור שטיינזלץ]. This careful approach was a direct correction of a previous, flawed attempt to transport the Ark from the house of Avinadav. During that earlier effort, the Ark was improperly placed on a cattle-drawn wagon [מצודת דוד] and moved without the use of its designated carrying poles [מלבי״ם].
To ensure the Ark was handled properly this time, the Levites took on the physical burden themselves. By bearing the weight upon their own shoulders, they avoided the past mistake of using a cart [מצודת דוד]. However, the Ark did not come into direct contact with their bodies, nor did it rest flat against their bare shoulders [רש״י]. Instead, the Levites placed garments over their shoulders, and upon these clothes, they rested carrying poles [מצודת דוד]. The actual staves of the Ark were then positioned inside these poles, allowing the Levites to lift and transport the sacred object securely and respectfully [רלב״ג].