How and why did Rome go from a Kingdom to a Republic?

Firstly it's important to note that there is basically no factual evidence to support the "history" of the Kingdom of Rome and it's transformation to a Republic.

In regards to your follow up question first, as it has direct relation to your original question, the first four of those six Roman kings personified a specific "Roman" virtue or performed specialised tasks. Romulus founded the city of Rome with Remus obviously, and was a warrior king who disappeared into the mists at the end of his reign. Numa Pompilius was a priest-king, forming a great deal of the religious rituals and traditional ceremonies. Tullus Hostilus built the first Senate House, and destroyed Alba Longa. Marcius built the first real bridge across the Tiber and codified how Rome would go to war so that it might remain in the right. As you can tell, these guys had very specific roles in their legends, Romulus especially who became deified as a god after his disappearance.

Additionally there is the question of there being only seven kings between Romulus and the Superbus. If we take the founding date of 753 BC for Rome and the formation of the Republic in 509 BC as factual, then each king had a fairly long reign, around 35 years on average, which is rather high, especially for the time period. The Romans themselves though were quite self aware about this, Livy remarking that the stories of the kings were "old tales with more of the charm of poetry than of sound historical record". Superbus for example is very much the tyrannical king, overthrown by the "will of the people" for the good of the city, by the nobility of course, because the mere plebs couldn't rule themselves.

As to why Rome became a Republic, well the legend goes that Superbus sent two of his sons to see the Pythia and find what the future held. They took with them Lucius Junius Brutus, the King's nephew who has survived the purges of Superbus by pretending to be an idiot (the cognomen Brutus means stupid in Latin). Once there they asked the oracle what it was they had been sent to ask (what the question was and the answer are not recorded), they decided to ask a follow up question, who would be King in Rome next. The oracle is supposed to have answered "He who shall be first to kiss his mother shall have supreme authority in Rome". The sons took the literal answer, their mother. Brutus however realised the oracle meant the earth, the mother of all and pretended to fall, pressing his lips to the ground.

Back they went to Italy, and on campaign at Ardea the officers of the Roman army got rather bored. Someone brought up the idea of going back to Rome to see what their wives were up to and one of the wives in question was a noblewoman named Lucretia. Sextus, the third son of the King rode back to her house a few days later and stayed the night, while there he raped her and afterwards she sent a message to her husband. Back he came, bringing Brutus with him, a friend he could trust. Lucretia names her rapist and promptly kills herself. Brutus at this point ditches his disguise and swears an oath to bring down Superbus and his family, heading to the Forum and rousing the masses in rebellion, led by himself and his fellow aristocratic conspirators. Superbus returns to find the gates barred to him, and withdraws to Etruria. Brutus is in command and he, along with his fellow conspirators establish a Republic, with a power sharing system similar to that of Sparta with its dual kings, except limited to one year terms.