tl;dr We don't know for sure because historical epidemiology is hard, but there are some good guesses.

"Decline" is the right word. Outbreaks of pestilence after the initial 1346-1351 continued to be catastrophic for centuries, but never quite reached the apocalyptic proportions of what we call the Black Death.

The first time around, the outbreak burned itself out as it killed off the people most susceptible, and (probably) the pest carriers of the germ. Paleoclimatologists have recently been arguing that the waves of plague arising in Asia, then travelling to Europe and the Near East, were the result of swelling gerbil populations in the wake of a wet spring plus warm summer. The usual thought from the European perspective is that these subsequent waves of pestilence from outside would coincide with times of increased vulnerability in the population.

Within Europe, particular outbreaks often seem to coincide with periods of notable increased vulnerability: a new generation without immunity from exposure; wars and temporarily swelling urban populations (like a German city when the Reichstag met in town--lots of localized plague outbreaks surrounding these) bringing people into closer contact and hurting sanitation; periods of famine and reduced access to food weakening people's resilience.

We can't say for sure why it started to taper off even before the widespread use of antibiotics, but good guesses track along the same lines. Better ideas of nutrition and fewer famines. More systematic approaches to sanitation and quarantine. (They had no real concept of germ theory, but knew that being near infected people often went poorly). The U.S. in particular got pretty good at turning away possibly plague-ridden ships, although we had our outbreaks and a few scary near misses.

I think the other commenter does an excellent job of describing the political and social reasons for this, but I think there is more here. We can look toward the evolution of the illness for answers as well. Mentioned above, as nutrition (often due to decreased populations) improved and quarantine procedures improved, the illness looked for other hosts. Traditionally, the vectors were parasites attached to vermin. Humans living in filth were around these parasites frequently. As hygiene, nutrition, and quarantine improved, the disease spread more to the vermin. By the time the Europeans are going to the New World, the rats on the ship had it (which is why a few cases appear in the US even today, mainly spread by prairie dogs), but by that point the disease that was killing people was small pox, a disease that needed no vector other than another sick person. Additionally, people may have changed. Like all disease, some people are more susceptible than others. Recent genetic testing has shown that Europeans have several genes which make them more immune to such bacterial infections (and may also show why they have stronger resistance to HIV/AIDS, though this is controversial still). Finally, there may be several diseases in play. While, like mentioned above, the disease evolved, it may also be a series of diseases. There is evidence, it could have been like the flu, with several strains at once. In fact, there was a study of five plague grave sites in 2003 that found no evidence of ‘bubonic plague.’ While another in 2011 found evidence that it was the bacteria, but it was so evolved as to be nearly unrecognizable.This has led several to conjecture that depending on the name (such as bubonic plague, black plague, black death) may even be different illnesses. Graham Twigg suggested it could be anything from bubonic plague to anthrax to a pneumonia based illness, that, like ebola, is such an efficient killer, it spreads to a population and kills the hosts so quick there is nothing left. Allowing the disease to die off. This was reinforced by Norman Cantor, but his book is frankly baffling, so I would not read it.