What does 探个究竟 actually mean?
Your understanding about 探 and 究竟 is quite right, I just add some additional information about 个.

个 has no specific meaning here, usually used between verb and complement for emphasizing, such as 笑个不停, 吃个饱.

So 探个究竟 doesn't just mean take a look or see what happens here, it describes how curious people are about the content in the barrel vividly.