There are a couple things wrong with your question but I'll answer the one I think you're asking. Let me know if you want sources; most of this is standard historical record stuff, although Suetonius' Life of Caesar and Life of Augustus are helpful. Livy, Sallust, and Plutarch are all relevant as well.

Roman history, as a republic, is the story of tension between two opposing forces. The opponents and the stakes changed over time, but the city was torn apart by conflict time and time again. Examples: patricians vs. plebeians, Romans vs. Italians, optimates vs. populares.

Around 100 BCE, this had escalated. The Gracchi, two members of a noble family, had just been killed for supporting land reform for the lower classes. Then Gaius Marius institutes his military reforms, the Social War devastates Italy, and Sulla and Marius have an extremely bloody Civil War. As Caesar, Pompey, and Crassus began to grow in power, they saw that an alliance of the three most powerful men should mean that if one got out of line, the other two would restrain him. We can see this in action as the years between Cataline's death and Crassus' death are quite peaceful. As soon as Crassus died, however, Caesar and Pompey almost immediately became at odds.

This was not a choice of the Roman people, as you seem to suggest, nor was it a cultural phenomenon. Rather, it was three politically savvy men attempting to avoid conflict.

The second triumvirate (Octavian, Antony, and Lepidus) was born from similar needs. Each man had a claim as Caesar's heir after his death: Octavian was his actual heir and politically brilliant, Antony inherited most of the client base and had been Caesar's magister equitum, and Lepidus... was actually somewhat irrelevant, but he held established political power at the time, and was included to prevent direct conflict by Antony and Octavian. Again, he was included by those two for that purpose, not by any vote. The triumvirates were not actual legal positions in the government, but rather organizations of the moment as circumstance demanded.

This system disappears until Diocletian introduced the Tetrarchy, or rule of four, at the end of the third century (although the Tetrarchy was a real system of government).

The answer to why they formed these triumvirates was to create a system of checks and balances at delicate times.