pushState is a way of going to a new page within a site without reloading everything. If you go to twitter/posts/1 to twitter/posts/2, then all the HTML and the twitter assets (images, css, javascript files, etc) will be reloaded. (OK, there's caching, but ... it can still be slow).

pushState lets you rewrite the bits of the page you want to change using javascript, without reloading anything inessential.

Hashtags do the same thing, because technically, hashtagged urls (twitter/posts#1 vs twitter/posts#2) are on the same page. It's a way of changing the URL in the address bar without technically changing the page.