the "analogue hole" refers to the idea that no matter how hard you try to make sure nobody can make illegal copies of videos, music, or text, there's always a "hole" in the protection that occurs when it's no longer digital. digital content can be protected - video files can be encrypted, music players can limit you to five devices - but analogue signals can't be. for example, itunes can disallow you from putting a song on more than five computers, but if you play the song through your speakers and record it, what you do with it is beyond apple's control.
one of the few remaining analogue outputs on a modern device is the audio jack. this is where you connect your headphones or speakers. your laptop only knows that something's plugged in. it doesn't know what's connected. you could be playing it through earphones or a massive speaker system and it wouldn't be able to tell. meanwhile, HDMI is a digital output format. it can tell if it's been plugged into a TV or a recording device, and can automatically disable output if you're recording it to make it harder for you to make copies of movies.
if the audio jack is replaced by USB-C or bluetooth, it becomes possible to tell what you're connecting to the laptop or phone. your phone might disable audio playback on anything but headphones to prevent you from hosting a public event with the music. it could also detect a recording device that you're using to (for example) make a copy of a song you're listening to on spotify and turn off the playback.
of course, at some point, digital needs to become analogue. humans can't watch electrical pulses, we need to see patterns of light. we can't listen to streams of 1s and 0s, we need vibrations in the air. this means that it's impossible to truly defeat the analogue hole - no matter what you do, someone can always just point a camera at their TV. it'll have worse quality, but it can never be stopped.
the analogue hole is one of many ways to circumvent DRM, or digital rights management. you can read more about it here: https://en.wikipedia.org/wiki/Analog_hole