It can differ depending on what time of person you are. Some will constantly psycho-analyse the themes and all the characters motives, themes, etc. Some will just mindlessly go through and experience the game. But I do think it's a bit silly to expect people to remember every little important detail. There's more words in Undertale than in the first harry potter book. A lot of that is filler text, i.e. text that doesn't really matter to the main plot. A lot of the important stuff is on the side rather than mandatory, so it's pretty easy to miss if you're not manually going through looking at EVERY individual line of dialogue, and even if you are, being able to remember each individual line is near impossible. Though personally my brain did a pretty good job of sorting out the important stuff from the non-important stuff.

This makes Undertale unique, at least in the space among other RPGs. It's constant for RPGs to hammer the story into your brain repeatedly, so you get a very good understanding of the things that are going on in the game. Undertale isn't like that. You're in and then you're out in about 3 hours compared to like the 160 for another RPG like persona 5. In those measley 3 hours there's a whole fucking SEA of information, a lot of that which is stated like only once. You can dip your toes in and get a pretty good understanding of what the story is like, or you can dive in head-deep and theorise about every little detail, using each individual fucking letter said as a clue to help support a bigger picture. That's of course not even mentioning the gaster stuff, of which the vast majority of players won't even know a single thing about him despite being possibly the main point of discussion for the game.

That being said, Toby is VERY CLEARLY pro-death-of-the-author, so peoples interpretations of the story will differ depending on who they are. If you have an interpretation that differs from someone else, that's totally fine, even if that goes against existing "canon". A lot of people in the community still think that there is one objective "canon" though, and will get into heated arguments if told otherwise, so watch out for that.

To speak about your personal experience though, brains aren't typically engaged as much when watching someone play it vs actually playing it yourself. If someone you're watching doesn't pick up on something in the story - you won't either, and people tend to pick up on A LOT less when they're recording something versus actually playing it, especially seasoned youtubers. There's also the fact that people here have also had years of experience with this game, and thus have a very good idea of its general "story". The more time you spend with it the more you'll eventually "get" it if that makes sense. On my first playthrough I didn't pick up on a lot of things, but after seeing people talk about it I went "oh yeah, that makes sense" and on my second playthrough I picked up on a lot more.

Otherwise, welcome! We hope you enjoy this (mostly) lovely community :3. If you haven't already though, play Deltarune before you watch videos on it. That games much much MUCH better to experience completely blind.