Great pictures, but even better editing! I've taken photos of both kinds of transit with pretty decent results, but I have no patience for editing - McCarthy's shots really pop in a way I don't know how to do.

It's actually not too hard to see the ISS or find a transit occurring in your area, there are great tools online to assist you.

https://spotthestation.nasa.gov/sightings lets you select your location and gives you sighting possibilities that can be up to a couple minutes long with the naked eye. These are fairly common and seem to occur in batches as the orbit of the ISS subtly changes day by day. The ISS orbital period is around 93 minutes, so sometimes you can catch multiple passes in a night - the issue is much more one of the evening being dark enough while the ISS is still receiving its own light from the sun.

To assist orientation I use the iOS app SkyView Lite, where you can search for the ISS and see its projected orbital path. The app was also useful for viewing the recent Saturn/Jupiter conjunction. I find with both spotthestation and SkyView, I have to look further above the horizon than either makes me expect.

https://transit-finder.com/ lets you find transits and near-transits in your area up to 30 days into the future. I've found their projections to be reliable and made about 10 attempts. After you calculate, you can click into "Show on Map" and then click the map to change your location to somewhere within the path, and you can choose exactly the chord that your transit will take. Andrew's "hours" searching for a perfect location is more likely just clicking around till you find a quiet street maybe on a hill - that's what I do. Clouds and fog day-of are a big factor in location selection, but actually knowing viable locations and being in the right ballpark is very easy.

As for equipment, I'm hauling around an expensive setup, but significantly cheaper options are doable and honestly should produce nearly as good of results.
I use a Canon 5D III or 6D II - the important thing to note is these are full-frame cameras, so in "close-up" astrophotography they're a good bit worse than crop-factor cameras like the much cheaper APS-C 1.6x crop Canon Rebel series. https://www.dpreview.com/forums/thread/4154657 has a lot more discussion of pixel density/relative magnification/crop factors but what matters for you is honestly just having the most megapixels.
For a lens I use the Canon 100-400mm IS with a 1.4x extender, giving me an effective 560mm zoom. For the cheap setup you could use a 70-300mm lens and save 90%+ on glass. A recent Rebel+70-300 lens kit can be found in the $400-500 range and would give you an effective 480mm zoom. I use a tripod and a wired trigger (such as, but not specifically this: https://shop.usa.canon.com/shop/en/catalog/remote-switch-rs-... ). This allows me to shoot without touching the camera, which will be extra important with a non-image stabilized lens.

The key is that a transit is a sub-second length event, and if you select one from transit-finder where the ISS in darkness is crossing a lit moon, you cannot see it coming. Most (or all, lmk if you know otherwise) automated trigger devices (hardware or software via apps) can't do sub-second intervals and the camera drive modes either are semi-automatic or shoot-until-full-buffer. Unless you can time your burst to the transit moment (which perhaps you can in a lit-ISS over non-full-Moon situation) you want to sustain your captures-per-second at a rate shorter than the transit duration (to guarantee you'll have at least one hit) and cover a much longer window than the event itself. Usually I will start shooting 60 seconds before the expected moment and overshoot by a minute or two, aiming for just over two shots per second. Test what your camera and memory card can actually sustain (card write speed can be a bottleneck here more than camera fps or buffer size) and use a card that has space for 500+ photos. Once you figure out your photo settings, you'll want to shoot at least one shot with the lens cap on so that later you may compare shots of the night sky with the sensor noise captured in the capped shot that might otherwise look like a false star or random blue pixel. Selecting the right camera settings is an art and a much longer discussion, but for now you just need to use Manual mode, manual focus (modern DSLR live view screen+magnified look is a godsend), aperture somewhere in f/8-16, and a shutter speed faster than 1/300.

This is a composite shot of the ISS passing the moon in March 2018 from lower right to upper left. I aligned several photos in photoshop by overlaying their moon positions, then made 10 vertical slices and selected the sharpest moon from the sequence and merged it all. There's very minor color saturation work (to separate the blue and orange a bit) and probably a minor sharpness tweak. EXIF data is included on this one if you want to compare settings.
https://www.flickr.com/photos/ohwaitnvm/40282783635

This is my first hit ever, during the blood moon of January 2018. The EXIF is missing from the upload but the settings were ƒ/10 1/2000 400ISO.
https://www.flickr.com/photos/ohwaitnvm/39289900094/in/album... The moon itself is about 850 pixels wide in the original image and the ISS is about 27 pixels, from which you can see a surprising amount of detail, though not quite as nice as the submission's pro work.

I have much less experience with solar transits (I've shot two, but editing is boooring and I haven't put them anywhere) but pointing your camera at the sun is not wise unless you have a solar filter such as https://www.bhphotovideo.com/c/product/1299066-REG/marumi_am... or I suppose the hydrogen-alpha solar telescope mentioned in the submission.

My takeaway is that maybe I need to pick up a 1.6x crop camera on the cheap soon.... And look into telescopes.