Final answer:
Samuel was the one who grew up at the shrine of Shiloh under Eli the priest. He played a key role during the era of Israel's transition from judges to a monarchy, paving the way for kings Saul, David, and Solomon.
Step-by-step explanation:
The individual who grew up at the shrine of Shiloh under the watchful eye of an old priest named Eli was Samuel. Samuel played a pivotal role in the transition of Israel from the period of judges to the institution of a monarchy. This transition is significant in the history of the Israelites as it ushered in the era of the united monarchy under kings like Saul, David, and Solomon, with Saul as the first king around 1030 BCE, David as his successor and the force against the Philistine threat, and Solomon building on his father's legacy by establishing the grand Temple of Jerusalem.