The idea behind it was that the browser default font size was 16px, so if you set the root font size for your document to 62.5%, then 1em = 10px, and designers who really, really wanted to work with pixels could use ems and pretend they were using pixels without doing complicated maths every time they wanted to set the size of something – they just needed to divide or multiply by ten.
The reason for not just using pixels was partly because it was drummed into everybody’s heads at the time that ems were better than pixels (which, in context, they were, but not when you pretend they are pixels), and partly because Internet Explorer at the time was unable to resize text set in pixels when you increased the text size, which was an accessibility failure.
The problem with this idea is that pixels and ems are fundamentally different types of unit. You can’t make them equivalent using any kind of calculation in CSS. Pixels are objective, whereas ems are based on user preferences. The browser might default to 16px font size, but users can pick something else. So if somebody had poor vision and increased their default font size; or if they had a small laptop screen and decreased their default font size, 1em * 62.5% != 10px, everything the designer set in ems was a different size than they intended, and a lot of their page layouts disintegrated into an unholy mess.
You can’t convert between ems and pixels in CSS. Any calculation like this is fundamentally broken from the start. Nobody should have ever used the 62.5% hack.