How can a manifold be Hausdorff and have an atlas where coordinate charts intersect? I'm sorry if this question seems "stupid", but I am having trouble visualizing a manifold that is Hausdorff with an atlas. Manifolds that are Hausdorff have non-intersecting neighborhoods and an atlas tells us a manifold is smooth. So let us say we are on a manifold, maybe the earth, if i have a atlas, I am able to transition between 1 coordinate chart to the other. Maybe outside of my country But then at the same time Hausdorff tells me something on this manifold needs to be non-intersecting. I don't understand what that is. Maybe i am going about this the wrong way?
 A: Hausdorff tells you that you can always find non-intersecting opens under certain conditions. It doesn't say that all opens are non-intersecting.
A: This is not a stupid question (not that there is no argument to be made for asking stupid questions). It took humanity a long time to formalize all this local-global stuff. Also as a disclaimer, there is a lot of fine print I ignore below.

To build on your example, consider the surface of the earth. We have atlasses (as in geography) with certain specified information (so an atlas could only contain the information of the borders of countries, or the distribution of sugarcane farms, or the distribution of infected people in the case of a pandemic). Considering an atlas in book form, we have different maps for different regions of the earth. As far as I remember some regions depicted in different pages overlap, so that the user always has some idea where the region depicted in a specific page is located. (I'll leave it to you to think about the modern day version of this.)
When you abstract away this idea into mathematics, being able to flip the pages of an atlas and being able to remember the maps of which two regions you want to use in tandem correspond to the transition cocycle you mentioned. Observe that just as there are different kinds of geographical atlasses that depict different information there are also different kinds of mathematical atlasses that depict different information. A smooth atlas like you mention is one that allows you to do calculus without worrying too much about running out of derivatives.
Let me also mention that a mathematical atlas bypasses any issues related to the so-called map-territory relation. As far as calculus is concerned, the surface of the earth is its depiction in an atlas (at least as a first approximation).

So far all I've mentioned was about the existence of an atlas. Let us add the Hausdorff assumption into the story. Hausdorffness guarantees like you said that given any two distinct points there are non-intersecting open sets separating them. This means that, returning to our example of a geographical atlas, given any two different locales, there is a page in the atlas that includes a depiction of the first one but does not include a depiction of any part of the second one. In other words you have the ability to localize to anywhere you wish. (I'll leave it to you to think of what this corresponds to in terms of geographical atlasses. In this case it's much easier to think of the modern day versions.) (As another little thought experiment, what does a "locale" mean here? One is not be able to do this with two neighboring countries, or two neighboring houses, for instance.)
This in turn allows you to localize to different regions, build a thing on each one of the regions without worrying about making a mess at regions you are not looking at, and then patch the thing you constructed at different locales up to get a global thing that works all across the earth. This is the so-called partition of unity (for which in general Hausdorffness is not sufficient but necessary, at least in classical contexts).

To sum up, both the notion of an atlas and Hausdorffness is precisations of the idea of being able to split a big thing into smaller things. An atlas tells you that as long as you know how to put down one small piece and take up another you have the whole big thing (and vice versa) (What does the previous paren mean?). Hausdorffness tells you that if you want to think about one specific part of the big thing and simultaneously ignore some other specific part of the big thing, there is a small enough piece you can pick up that does the job. This (under certain conditions) means that if you want to build a thing on the big thing you can build things on each small piece and put them together. (For this last paragraph it might be useful to think of a puzzle of the earth, with somehow the sizes of each puzzle piece changing at your will.)

Finally here are two natural exercises that I hope will be useful to think about:
Exercise 1: What is an example of a topological space that is Hausdorff but does not have an atlas (in the sense we consider here)?
Exercise 2: What is an example of a topological space that is not Hausdorff but does have an atlas?
