It all honestly seemed a lot easier than what has come after.
(The funny thing is that, in modern browsers this is no longer true - you actually can use CSS to restructure table layouts now. But, the world has already moved on.)
Though there's a broader answer, which is that generally as web design has modernized, so too have ideas of "semantic" markup. Semantically, table is supposed to be used to display data. HTML which is used for its proper semantic purpose is easier for screen readers, improving accessibility, and easier for search crawlers to make sense of the content, improving SEO. So a lot of old HTML tricks fell out of use when HTML5 started becoming popular.
Table-based layout was already falling out of favour before this happened. Even when you had fixed sites that were predominantly desktop, divs and spans (coupled with CSS) made life a lot easier to put things where you wanted them and lay out a site faster. It was also cleaner when reading the markup.
From what I remember, the move away from tables happened around the same time external CSS took off and you could partition your site with reusable styling instead of framesets or tables.
Responsive design then sped up the process a few years later.
It's funny looking back at this SO question from 2009, and its answers: https://stackoverflow.com/questions/426253/why-use-tables-to...
There was definitely an ongoing culture war between people who felt like using tables was not "proper", and people who just wanted their site layout to work consistently in IE6.
The layouts people were trying to create with frames weren’t bad, but browser behaviour around framesets isn’t good. I didn’t really have a good enough Internet connection at the time, but looking at them now I do agree that frames break linking too hard and shouldn’t be used.
I think frames are an artifact of the time when people thought of HTML as a desktop publishing markup language. It has evolved into a generalized UI description language.
Ironically people moved to divs and spans, which when combined with the old box model issues, made the problem even worse.
The semantic web elements in HTML5 was an attempt to fix those issues.
The reason we moved away from it was that after a point, the amount of legwork to implement relatively simple behaviours became astronomical. Whether or not the current state of matters is "better", who's to say. But things are certainly much easier now.
Besides, you still can use those old modes if you want to :)