dude the guy you're listening to has autism. he has no idea what he's trying to explain.
a topological space is simply a set where we can talk about things being "attached to" or "near" other things. an example which everyone learns first is called a metric space, which is just a set where you can measure the distance between any two points. there are topological spaces which are not metric spaces, but they tend to be pretty weird.
a topological space doesn't really care about shape, just about the nearness. for that reason people call topology "rubber sheet geometry" because you can twist and bend a topological space, and as long as you don't tear it or glue it together it doesn't really change.
so just think about it like some weird wavy surfacey/curvy thing.
now, what's homology good for? turns out it's really hard to tell how many holes a topological space has. why is it that we cant bend/twist a donut into a sphere? cause the donut has a hole. but how do we describe the hole mathematically?
one way is homotopy, which is probably a lot easier to understand than homology.
but the other way is homology. homology lets you slap a simplex into your space and ask what simplices it can be written as. well, if you slap a triangle onto a donut which goes around the hole, then try to break it up into other triangles, one of these triangles still has to go around the hole (there's a proof needed there). but on a sphere, if you slap a triangle on, you can just pull all the vertices together and make it super small til the triangle is a single point. this is what homology is distinguishing (via a "quotient from a boundary map").
i'll be honest with you, it takes a while to learn all the math to make the formal description make sense. but an intuition is important as a first step.