Why did the Emperors leave Rome?

By the third century, Rome was basically irrelevant to the Empire.

(1) Rome and Italy was no longer the rich center of the Empire -- the provinces had long since equalled Italy in most respects.

(2) The army's officers were provincials, because the main challengers to the first and second century emperors came from the Roman aristocracy. As a consequence, the Roman (and Italian) aristocracy slowly ceased to provide the bulk of the army's officer corps.

(3) Because it was recruited from and supported by the provinces and its officers were almost entirely provincials. Romans and even Italians were relatively scarce. The army was no longer the army of the city of Rome in anything but name.

(4) The Emperor was no longer the head of the Roman State, he was head of the Roman Army -- the State had become merely the Army's supply mechanism. The emperors knew that.

(5) Rome was not a healthy place to be Emperor. The Roman Senate and its associated aristocrats still thought they ought to be running things and the Praetorian Guard still liked the profit potential of making and breaking emperors. If the Emperor, the Emperor's court, and most of the power resides elsewhere, these destabilizing influences were minimized.

(6) As noted in other answers, the army's main focus was the frontiers, particularly the Danube-Rhine frontier. Northern Italy was several day's travel (and one mountain range) closer giving the emperor much better control.

(7) Rome was an economic basket case with a huge population, a significant fraction of which was fed at public expense. An emperor in Rome had the Roman mob to deal with. No fun at all.

(8) As the emperors left Rome, so in time did the people. Its population dropped by at least half during the late Empire, making Rome even less relevant.

By the late Empire, the City of Rome was an (expensive) historical monument funded because, well, we have always funded it. It was still symbolic, but no longer the center or source of the Army, the administration or trade and was essentially a troublesome parasite. No one wanted to toss it aside, but its centrality was nibbled away a little bit at a time.

The emperors leaving Rome is an exceptionally clear example of broad historical trends driving historical development -- no Great Men had any particular impact on the process.

Wikipedia has the cursory answer

The transfer was made partly for defensive purposes: Ravenna was surrounded by swamps and marshes, and was perceived to be easily defensible (although in fact the city fell to opposing forces numerous times in its history); it is also likely that the move to Ravenna was due to the city's port and good sea-borne connections to the Eastern Roman Empire.

A full answer would have to address a few more issues

• Rome was irrelevant. The Roman Senate continued to behave as though it had some function in ruling the Empire, but the Emperors were increasingly autocrats.

• During the Year of the Five Emperors, and Crisis of the Third Century, the legitimacy of the Emperor effectively changed from the consent of the Senate to the backing of the Legions. Some Emperors continued to go through the forms of requesting Senatorial approval, but it wasn't required.

• Diocletian was insulted by the lack of respect shown to him in Rome (I think this is accepted, but not fully proven). He and his successors saw no reason to go back to a town that didn't realize that the Emperor's visit was a privilege granted to Rome, not the other way around.

• @FelixGoldberg is quite right to point out that the Emperor frequently resided elsewhere for his entire reign; vote up his comment