Ah, etymological questions about Latin! I can certainly help here. The word from which all these words are derived can be found here: Lewis and Short Dictionary - occido.

You can see that the word means primarily 'to fall down'. 'Occident' is a loaned form not used in Latin itself, derived from the verb form 'occidens', and occasum is also a form of the same verb. In the context of the sun, the first two literally mean '(the sun) falling down' and the second '(the sun) fallen down'. So you can see how this would apply to the direction West and the sun. If you look at the sun directly without any knowledge about astronomy, it looks like it 'falls down' in the same place every day. The meaning of 'death' or 'dying', which is also in the dictionary entry, doesn't really come into play here though it's a natural extension of the basic meaning of the verb. Think like, 'the soldiers who fell in war' for an analogous English usage. Here it's just a reference to the setting of the sun below the horizon.

The opposite term 'orient' comes from this Latin verb: Lewis and Short Dictionary - orior. It means 'to rise up'. So the two terms complement each other straightforwardly, in the same way we use the terms 'rise' and 'set' in English to describe these phenomena. The 'orient' is where the sun rises, the 'occident' where it falls or sets.

So, as you can tell, these words don't have to do with any particular cultural significance of the Atlantic ocean, besides that being the direction from Europe where the sun sets.

Edit: added some more fun info about the word 'orient'

Edit2: addressed the cultural significance of the Atlantic

Edit3: caught something I should have phrased more precisely

Follow up, is Occitanie somehow mixed up with this?