Why did an emperor build a new Rome on the Bosporus?

Because Rome had stopped being a good place to run an empire from. The emperor Constantine decided that Rome was too far from the frontiers, and so too far from the armies and the imperial courts, and that it offered an undesirable playground for disaffected politicians. So he founded a new capital on the site of an old Greek city called Byzantium, first calling it Nova Roma, "New Rome". In 330 it became the new eastern capital of the empire, and it soon became more widely known by a name in honour of its founder: Constantinople, "city of Constantine".

It might have seemed unthinkable to move the capital, since Rome had held the title for over a thousand years. But Constantine saw what the site offered: a place where an emperor could sit, readily defended, with easy access to the Danube and Euphrates frontiers, his court supplied from the workshops of Roman Asia and his treasuries filled by the wealthiest provinces.

The geography did much of the work. The site lay astride the land route from Europe to Asia and the seaway from the Black Sea to the Mediterranean, and the Golden Horn gave it an excellent, spacious harbour. Its position between the Golden Horn and the Sea of Marmara also reduced the land area that needed defensive walls.

Kings had noticed the spot long before. In 512 BC the Persian king Darius I annexed the city because it stood at the narrowest point of the Bosphorus, the best place to build a pontoon bridge into Europe. Even the Roman emperor Septimius Severus, who razed the city in a civil war, rebuilt it afterwards. Constantine made it a capital, and it stayed one for a very long time: after the western parts of the empire were lost in the 5th century, Constantinople was the sole capital.