In general, they "prove" something by applying parts (or all) of their postulations to real world experiments. Most of our understanding of the extreme universe is simply a model. The model is used to describe the actions of the observations; they don't always match, but that's part of science.

quote:
We’ve never been far enough into space to see and touch stuff.

Like above, it is mostly just a model and they try to make that model fit as closely as possible.

quote:
even with our most advanced cameras and telescopes, the photos have to be doctored with color or else they look like they’re taken with a potato.

Yes and no. The vast majority of images you see published are not images that you could "see" with your normal vision. Most of the astronomical images are computer generated images of radio waves. They either "doctor" the image to give a visual representation of what the telescope (usually a big radio antenna) detects, or the image is "doctored" to add false color. Sometimes the images are augmented to make for a pretty picture, sometimes to show different elements of the image in different colors. So, more often than not, the picture you see in National Geographic isn't a picture that just been taken with a really fancy camera, but it is a real representation of observed data.

As mentioned most observations are made with radio telescopes. These "telescopes" detect different radio-waves (electromagnetic waves) and from those different waves we can determine different elements that comprise the object we're looking at. Then we can layer it further by looking at velocity and direction of the objects.

quote:
Are these and other breakthroughs purely theoretical?

With the exception of the most "out there" postulates, the breakthroughs come from actual observations, or from new math/physics that explains observations. Many can be studied with local experiments - physics is the same everywhere in the universe (probably

).

To "test" the theories:
-We can study high energy collisions, like those from supernova, by using particle accelerators like the Fermi Lab and the LHC at CERN.
-We can study ultra high heat and pressures at labs like the National Ignition Facility in California.

To "see" what's happening in the extreme universe:
-We can study data from extremely large detectors like LIGO.
-We can also study data from real world experiments that show the "unseeable" universe like the Double Slit Experiment or the observation of the negative pressure.

We can also see the real world effects of some of the "crazy" physics in things like your phone screen or computer processors (quantum tunnelling).

ETA - I'm trying to answer you and pay attention to the professor at the same time, so you'll have to excuse my errors.