The source of Vonnegut's statistic is an especially unfortunate one, in that it comes from David Irving's The Destruction of Dresden. Irving of course would become notorious in later years as a Holocaust denier, and while at the time of Vonnegut's writing of Slaughterhouse Irving had not publicly espoused those beliefs, as far as I am aware Vonnegut never addressed this in later years (Irving's trial occurred in 2000 and Vonnegut died in 2007).

In fact, the current estimates of 25,000 dead (per research released in 2008 by the Dresden City Council) very closely match causality figures provided by local authorities in the immediate aftermath of the attack. Irving based his higher estimates off of literal fake news, namely the Tagesbefehl 47 put out by the Reich Ministry of Propaganda under Josef Goebbels that claimed to be documentary evidence showing 250,000 deaths (he moved the decimal point). This was the figure Goebbels provided to Swedish journalists at a press conference after the bombing in an attempt to embarrass the Allies.

Irving would later admit in 1977 that TB 47 was a forgery, but in various editions of his book remained both vague about his numbers, while still inflating them greatly. He would write things like "from a minimum of 100,000 up to 250,000" to sound reasonable, and in later editions, when the authenticity of his documentation was challenged, he would retreat back to a 100,000 estimate.

Irving's estimates even at the time of his writing were higher than the casualty estimates provided by other historians or governmental bodies, which assumed the initial casualty count to be low, but put the total death toll in a 35,000 to 50,000 range.

It is a great problem that such a well known work of fiction that takes the Bombing of Dresden as part of its subject matter cites by name such a controversial and unreliable historical source.

Richard Evans. Telling Lies About Hitler: The Holocaust, History and David Irving Trial

First off, it's important to remember that war casualties are always notoriously hard to find out, and all casualty numbers for every battle are normally calculated to an estimate based on head and body counts, etc. by the same militaries that may also have an incentive to skew said numbers for propaganda purposes. Even if they do not of course, it's simply an impossible job to get down to an exact number.

Secondly, Dresden was from the moment it happened, used as a Cause célèbre by the Nazi regime at the time. This is important to your question, as it is entirely possible that said propaganda stuck with Vonnegut, intentionally or no, considering he was in Dresden during the attacks, and was recruited to assist with the cleanup. Because of this, he almost certainly would have been privy to the German response via the press to the attacks, as well as obviously the public reaction of the Germans in Dresden proper. The Nazi regime told the press to hype up the casualty numbers, and so reports from anywhere between 200,000 to 500,000 were reported in some cases, none of which was supported by actual fact. Even at the time, officials within Dresden reported something closer to 25,000.

That number has been researched since, both by the modern German government and other independent sources and found to be fairly accurate. However, as there still exists people sympathetic to the Nazi cause, there still remain people sympathetic to the idea of Dresden as a cause célèbre. The modern Neo-Nazi movement, both in Germany and elsewhere, have kept the myth of the "Holocaust of Bombs" alive, inflating the numbers based on obviously politicized sources within the Nazi government.

And it's at this crossroads that Kurt Vonnegut's number comes from. Vonnegut of course lived through the fire bombing, and was recruited (as he was a POW at the time) to assist in the cleanup of bodies/searching for those who may still be alive. He recalls the damage in pretty grave detail in his writings, and has called it specifically the "Dresden atrocity" in a number of accounts. Around the same time that he was writing "Slaughterhouse-Five," David Irving, an avowed Holocaust denier, wrote a book on the bombing of Dresden in which he utilizes information and figures that were inflated (even at David Irving's own admission at this point) to give the 135,000 dead number. Vonnegut used that research at the time in his own telling of it, potentially for similar reasons: to portray the attack as an unnecessary atrocity.

(EDIT: After a bit of further research I came across a more direct connection than I had even thought between Vonnegut's number and the research done by David Irving at the time, and changed my response to match. Apologies for the change if you saw the original piece.)

Further Reading:

http://lipstadt.blogspot.com/2007/04/kurt-vonnegut-and-david-irving-in.html - Deborah Lipstadt, mentioning specifically a connection between David Irving (who she sued for libel in a famous case relating to his holocaust denial) and Vonnegut's number in Slaughterhouse-Five

"The Destruction of Dresden" by David Irving, if only to learn where the falsified information/number originally came from and as a resource to understand how misinformation spreads

https://www.spiegel.de/international/60-years-after-the-bombing-of-dresden-a-war-of-words-a-339833.html - An article from "Der Spiegel" that goes into the modern Neo-Nazi response to, and continued mythologizing of the Dresden bombings.

"Dresden: Tuesday 13, February 1945" by Frederick Taylor