This passage (among many others) emphasizes the importance of doing right no matter how inconvenient it may be. King Saul certainly found himself in circumstances that were not convenient. He had a standing army of a few thousand, but they had no weapons. And, his people were terrified of their enemies, the Philistines. The Philistine's army was massive and they were very well equipped. Saul's people were scarce, scattered and running scared (I Samuel 13:6). Even the few that were following him were useless militarily (I Samuel 13:7). And, on top of all of that, the prophet Samuel was late to the appointed time of sacrifice. Saul desperately needed something to happen to unify his followers and to rejuvenate their confidence. So, he finally took matters into his own hands. I Samuel 13:9 tells us the sad news. Saul did something that he was not qualified to do. He was not a priest... but he acted like he was. He offered the sacrifice himself.
Isn't it bitterly ironic that Saul had no sooner finished the sacrifice when Samuel showed up. If only he had waited a little longer, right? But no, if he hadn't failed then and there, he would have failed in another way somewhere else. His heart was drifting from God already... this just happened to be the way that his character flaw was manifested. If you examine Saul's response to Samuel, you'll discover that Saul had a hard time admitting his faults and failures even when he was caught red-handed. Anyway, Saul's kingdom was doomed. While he did still have 38 years remaining to rule as king, God had already picked his replacement (I Samuel 13:14).