I'm sure that there are plenty of people out there who have no idea what a plague doctor is/was, so I'll start with an explanation.
Despite what we now consider to be their relatively fearsome appearance, they were originally just doctors, and they have existed for as long as there have been plagues. Our first record of plague doctors dates back to around the year 500 AD, well before the germ theory of disease. These doctors would be contracted by the government to treat everyone with the plague, rich or poor, and as a result, their lifespans were rather short. This is unsurprising, considering that any protective measures they took were almost certainly misguided or totally ineffective. Generally speaking, 90% of plague doctors died from the diseases they were trying to treat.
In 1619, Charles de L'Orme created what we now consider to be the distinctive "look" of a plague doctor. Here's a picture of a plague doctor from 1656:
The body was covered, head to toe, in an attempt to keep the disease at bay. In fact, the stick that the doctor above carries was used to prod patients so that they didn't actually have to be touched.
Meanwhile, the most distinctive part of the costume, the mask, was created in order to hold incense or herbs that ostensibly kept out the "bad air", and thus prevented the plague doctor from getting sick. Ideally, at least, as it didn't really work in practice. The end effect, of course, was somewhat terrifying, but it's important to note that underneath the strange costumes were men who were trying to cure the ill. Their pay was high (nearly four times higher than a regular doctor, by some accounts), but their mortality rate was also high.
We look at the masks as some sort of adornment, but for the plague doctor, they were strictly functional and highly unpleasant. In fact, wearing the plague doctor outfit must have been an absolutely horrible experience, but they did it anyway.
Were it not for their fearsome appearance, plague doctors might have been considered heroes. Granted, many of them were scam artists trying to cash in on the high pay, but some of them were genuinely trying to help.
So why is Steampunk absolutely inundated with plague doctors? On the surface, the plague doctor would seem to be ideologically opposite to Steampunk.
First of all, with the popularization of the germ theory of disease in the mid-1800's, plague doctors became more or less a thing of the past. Or at least, the plague doctor costume became very outdated. So plague doctors don't really fit into the time period that's often associated with Steampunk.
Second, plague doctors in many ways represent superstition over science. Breathing incense to keep away the sickness? Seems rather silly to a modern audience, and much more like nonsense than science.
However, it's important to remember that for the time period in which the plague doctor costume was invented, that was science at its finest. These people weren't shaking sticks and praying for cures, they were actively trying to treat patients using the latest medical science at their disposal. Granted, that was usually something like, "drink this mixture of herbs that I found in my yard", but it was still something.
Another thing that lends to their credibility as both scientists and objects of terror is that plague doctors were given permission to perform autopsies on plague victims in the hope of curing the disease. That may not sound like much right now, but at the time, autopsies were considered blasphemous, evil, terrible, etc. So in all likelihood, they were doing some solid work toward our understanding of anatomy, if nothing else.
Looked at from that perspective, you can almost see plague doctors as champions of science, or maybe martyrs for science would be more appropriate, given their mortality rate.
Another thing that Steampunks like are costumes, and few historically-accurate costumes are as great as the plague doctor's. I mean, come on... Plague doctors look like terrifying birds, and they poked people with sticks. What's not to love about that?
So is it really any wonder that the plague doctor has captured the imaginations of Steampunks? Men of science who were viewed with distrust and suspicion by the public, and who are now an obscure and forgotten part of history? No, you're right, that doesn't sound like something Steampunks would like at all.