Preparing the Passover sacrifice required a highly organized and specific division of labor between the priests and the Levites. The Levites took on the initial task of slaughtering the animals. This arrangement was appropriate because the act of slaughtering a sacrifice is entirely valid even when performed by individuals who are not priests [מצודת דוד].
Immediately following the slaughter, the priests stepped in to handle the blood. The primary approach among commentators is that the priests stood ready with their vessels to receive the blood directly from the hands of the slaughtering Levites. To ensure the process was seamless and the blood was immediately ready to be sprinkled, some explain that the priests actually placed the collection basins into the hands of the Levites ahead of time [מצודת דוד].
Once the blood was collected, the Levites resumed their duties by removing the skins from the animals. A clear and complete picture of this shared work emerges: the Levites managed the physical labor of slaughtering and skinning the animals, while the priests focused entirely on receiving and sprinkling the blood [מלבי״ם].