What does the hadith of the three men trapped in the cave teach about sincerity?

The hadith of the three men trapped in the cave emphasises the importance of sincerity.

Abdullah bin Umar may Allah be pleased with him narrated that the Beloved Prophet ﷺ said, "Three men from among those who were (from a nation) before you, set out together till they reached a cave at night and entered it. A big rock rolled down the mountain and closed the mouth of the cave. They said (to each other), 'Nothing could save you from this rock but to invoke Allah by giving reference to the righteous deed which you have done (for Allah's sake only).' So, one of them said, 'O Allah! I had old parents and I never provided my family (wife, children etc.) with milk before them. One day, by chance I was delayed, and I came late (at night) while they had slept. I milked the sheep for them and took the milk to them, but I found them sleeping. I disliked providing my family with the milk before them. I waited for them, and the bowl of milk was in my hand and I kept on waiting for them to get up till the day dawned. Then, they got up and drank the milk. O Allah! If I did that for Your Sake only, please relieve us from our critical situation caused by this rock.' So, the rock shifted a little, but they could not get out."

The Prophet ﷺ added, "The second man said, 'O Allah! I had a cousin who was the dearest of all people to me and I wanted to have sexual relations with her, but she refused. Later, she had a hard time in a famine year and came to me. I gave her one-hundred-and-twenty Dinars on the condition that she would not resist my desire and she agreed. When I was about to fulfill my desire, she said: "It is illegal for you to outrage my chastity except by legitimate marriage." So, I thought it a sin to have sexual intercourse with her and left her though she was the dearest of all the people to me and I also left the gold which I had given her. O Allah! If I did that for Your Sake only, please relieve us from the present calamity.' So, the rock shifted a little more but still they could not get out from there."

The Prophet ﷺ added, "Then, the third man said, 'O Allah! I employed few laborers and paid them their wages with the exception of one man who did not take his wages and went away. I invested his wages and got much property thereby. Later, he came and said to me: "O Allah's slave! Pay me my wages." I said to him: "All the camels, cows, sheep and slaves you see, are yours." He said: "O Allah's slave! Don't make fun of me." I said: "I am not mocking at you." So, he took all the herd and drove them away and left nothing. O Allah! If I did that for Your Sake only, please relieve us from the present suffering.' So, that rock shifted completely and they got out." [Sahih al-Bukhari 2272]