Note that what's called a 'cable' is not just one single transmission line, it's a bundle of tens of fiber optic cables (strands of transparent plastic). They work by laser light that's shone through them and flickers in an extremely high frequency to deliver gigabits or even terabits of data per second per fiber. Modern versions even use multiple lasers with different colors to transmit multiple streams of data through the same fiber. As the plastic isn't perfect, after a distance of a few hundred miles they need a receiver (an electronic 'eye' to see the laser signals) connected to a new set of lasers to 'repeat' the data for the next few hundred miles. These are powered by separate power lines in the cable, with a very high voltage to be able to travel such a long distance without losing too much power in the cable itself.

As a sidenote, specifically for repetitive stuff like cat videos or your latest Netflix series, video websites and streaming services use local servers in every part of the world to serve the content to the local users. They in turn request the data from centralised servers, but that only has to be done once per chunk of video, saving a lot of bandwith. This is called 'caching' and removes the need for most of the basic internet stuff to have to cross the Atlantic all the time. It's also noticeable when you watch more obscure videos or listen to less popular music that it sometimes takes a few extra seconds to start playback. This is caused by the local server not having the content ready and first has to get it from the central server(s).