What does 探个究竟 actually mean? On the lid of the barrel（桶）  it says '不可偷看‘ 

I have the sentence：人很好奇，走过来弯下腰把脑袋伸到桶里探个究竟。

探 = explore, 究竟 = outcome, result

探个究竟 = 'take a look' or 'see what happens' or both OK?
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.