Before embarking on a military campaign, it was necessary to seek divine favor through prayer and sacrifice. As the threat of war looms, Saul feels the urgent need to secure God's help and takes the initiative to perform the offerings himself [מלבי״ם, ביאור שטיינזלץ].
Samuel had previously instructed Saul to wait a full seven days for him to arrive and lead the service. When the morning of the seventh day dawns and Samuel has not yet appeared, Saul takes action, ordering the pre-prepared burnt offerings and peace offerings to be brought to him [רד״ק]. Saul justifies his decision by reasoning that Samuel's strict order to wait applied only to launching the actual battle without permission, not to the preliminary act of offering a sacrifice to please God [מלבי״ם].
Ultimately, Saul's failure stems purely from impatience. He was expected to wait until the very end of the seventh day, toward evening. Had he held out just a little longer, he would have met this requirement, as Samuel arrives immediately after the act is completed [רד״ק].
Despite his impatience, commentators agree that Saul does not sin by assuming the role of a priest. During this era, the central Temple had not yet been built, making it entirely permissible to use private altars. Under these conditions, any individual, even those outside the priestly family, was fully permitted to offer sacrifices.