As it's used in the first chapter of Jerome K. Jerome's Three Men in a Boat, "seedy" means "unwell" or "poor," often as if you've been eating or drinking way too much:
We were all feeling seedy, and we were getting quite nervous about it. Harris said he felt such extraordinary fits of giddiness come over him at times, that he hardly knew what he was doing; and then George said that HE had fits of giddiness too, and hardly knew what HE was doing. With me, it was my liver that was out of order. I knew it was my liver that was out of order, because I had just been reading a patent liver-pill circular, in which were detailed the various symptoms by which a man could tell when his liver was out of order. I had them all.
As you can see in that paragraph from the story, the men are feeling "seedy," specifically meaning that they are feeling dizzy and otherwise sickly.
The Oxford English Dictionary actually cites Jerome's use of this "seedy" as one example of the word's meaning of "unwell, poorly, 'not up to the mark,' specifically as a result of excessive eating or drinking." The idea is that the seedy person is looking pretty shabby, like "a flowering plant that has run to seed," in the words of the OED. Imagine a flower that's already shed its seeds. It's becoming less colorful, less strong, and closer to death.
So, "seedy" is the perfect word for the men in this story. Even though most of their illnesses are imagined, they probably really do look all droopy, tired, and sallow as a result of how they constantly sit around doing nothing but trying to look sick.
In addition to describing what unhealthy people look like, "seedy" also describes what run-down, dirty places look like, and how their reputations are similarly icky. A seedy city street, for example, is clogged with garbage and full of cracks. By extension, a seedy restaurant or a seedy politician has a lowdown, dirty, slimy reputation.
It's good to know, however, that "seedy" can also simply mean "full of seeds," as in "Wow, this watermelon is delicious, but it sure is seedy."