Getting clear beer isn't really all that important. But if in the case above it was a brew that used corn as an adjunct to the mash I think that leads to less color in the beer which makes it seem extra clear. Also, this beer was not super cold, that is cheating a little bit.
In terms of process, make sure that you have a good strong boil, don't be afraid to use a whrirfloc tablet in the brew. Take good care of your yeast. I always make a starter. Ferment at appropriate temps. Chilling the beer quickly also helps a ton. I strongly prefer counterflow chillers and plate chillers to the concept of immersion chilling. I believe the quicker the beer loses temp the better cold break one can get. A plate or conter-flow only needs a few seconds of contact to perform cooling, whereas the immersion chiller may need 10-30 minutes.
After that It is just a matter of patience. I wait for a solid yeast cake to be visible in the bottom of the carboy, then I rack to secondary for 2 more weeks, longer for a big beer. After that, in bottles or keg for about 2 weeks and they should be optimal. In general, I expect to drink a beer 6 weeks to 2 months after brewing.
Don't stress the details too much, one thing at a time and keep having fun.
Sanitation is still the most critical detail of all. Or maybe the yeast. Both really.