To add to the last bit: part of the post processing is taking advantage of the fact that celestial objects move so slowly they might as well be static.

Now, when cameras take pictures, sometimes there are tiny "blips" in the photo (for a myriad of reasons) which creates noise in the photo (slight colour irregularities; you'll see a lot of noise in low-light photography with your cell phone, for example).

But since objects in space don't really move at all over days/weeks/months/years (depending on what we're talking about) you can take as many photos as you want.

Aside: Even crazy long exposure photographs. With a computerized motor that matches the rotation of the Earth, you can take photos with exposure times of hours.

Then you put all those images into computer software that compares them all. If most of the images agree on a pixel, then it's probably "correct", so any "blips" of noise can be eliminated (or at least drastically reduced).

You can also use different lenses and filters to only look at specific frequencies of light one by one, then combine them all using computers. And do all the exposure and stacking techniques above to make the individual spectrums more clear.

Combine the above, and you can get incredibly clear photos, even with a backyard telescope (with the right tools and a lot of patience!)

Another thing I didn't see above is using multiple radio telescopes that are very far apart. Looking at some lower frequencies of electromagnetic radiation, they can effectively combine their images to make a "virtual" telescope that's way bigger than they are.