Scientists often make claims about what exists. A compelling and powerful argument for why this OK is that you can check their claims for yourself. However, most people (myself included), often don't have the time, the means, or the interest. We trust that other people have done the things that they claim. Indeed we rely on it, and have little reason to doubt them (cars work; computers work; telephones work).
However, when I haven't done or seen something for myself, I find that it's easier for me to maintain some shred of doubt or incredulity. I only reach the point of overwhelming certainty in a story when I check it for myself.
A classic story in astronomy, and all of science, concerns Galileo Galilei, turning a telescope he built from dutch lens-maker's plans towards Jupiter. This morning, I did the same!
This entailed waking up at ~5am to get to the telescope by 5:20am. This was the hardest part. (Admittedly, getting to the point where I have easy access to a 12" telescope in dark skies is harder, a priori). From there, Jupiter is BRIGHT (V~-2) -- way brighter than anything in the sky. Using the 12" I pointed right at it (& aligned the scope), and then looked. The telescope was way out of focus, so I brought it back in, and then BOOM.
Big disk. 4 bright points around it. Three on the upper right, one to the bottom left. Disk has two easily resolved dark bands. (In the same plane, roughly, as the 4 moons). The 4 points: Io, Europa, Ganymede, Callisto. (Which is which? I'm not sure yet. It will require more observing).
One looks like it's a bit inclined with respect to the plane of the other three (however this is a trick of forced perspective).
Jupiter's disk is massive. The magnification we had on the scope was perfect for the job.
(Although I still need to learn how to do better filters and magnifications. And how to take better pictures)
Pics or it didn't happen; put my camera phone to the eyepiece. Some day soon: better cameras, less blur. Pictures below are with no filter, and with a bit of recoloring to remove glare. The bands aren't visible, but the moons and planet are.
2016/11/17 update: ID'd the moons, courtesy of Mark Showalter's website.