There's no real "secret sauce", it's mostly just "do things smart". For example, while MathJax tends to do a bunch of calculations about exact positions of where different characters should go, KaTeX manages to get away with offloading most of that into the CSS, so the native CSS renderer can handle it.
MathJax also does strange things involving inserting elements into the DOM to look at metrics that it uses in its calculations, then removing them, which takes a lot of time. We make sure that the same input is rendered the same every time, and try to make sure it looks good everywhere (there are a couple cases where we aren't the best at this, mostly because browser font rendering isn't great).
We also support less environments than MathJax, for example we only support HTML+CSS output (for now, though MathML is in the works for accessibility reasons), whereas MathJax supports HTML+CSS, SVG, MathML, and older browsers like IE 6. This means it can't rely on things like good CSS rendering like we do. It's a worthwhile tradeoff for us, because we can make ours so much faster, but it is a tradeoff.
MathJax also does strange things involving inserting elements into the DOM to look at metrics that it uses in its calculations, then removing them, which takes a lot of time. We make sure that the same input is rendered the same every time, and try to make sure it looks good everywhere (there are a couple cases where we aren't the best at this, mostly because browser font rendering isn't great).
We also support less environments than MathJax, for example we only support HTML+CSS output (for now, though MathML is in the works for accessibility reasons), whereas MathJax supports HTML+CSS, SVG, MathML, and older browsers like IE 6. This means it can't rely on things like good CSS rendering like we do. It's a worthwhile tradeoff for us, because we can make ours so much faster, but it is a tradeoff.