Basically, the study of knots is the study of how the simplest 1 dimensional thing (the circle), can sit in 3-dimensional space. And it turns out that even this "simple" case is incredibly rich and difficult. So that's a reason to expect knot theory to be an inherently interesting thing to study. So topologists, and especially topologists specialising in 3-dimensional objects were always interested in knots.
In the 1980s, Vaughan Jones discovered the Jones polynomial, which is a property of knots which remarkably turned out to have deep connections to all sorts of things including quantum field theory! This led to 3 decades and counting of intense study into the relationship between knots and fundamental physics. I'd like to say more, but I'm knot really qualified to speak about the connections to other fields. So that's basically the tl;dr of why so many people care about knots!
In the 1980s, Vaughan Jones discovered the Jones polynomial, which is a property of knots which remarkably turned out to have deep connections to all sorts of things including quantum field theory! This led to 3 decades and counting of intense study into the relationship between knots and fundamental physics. I'd like to say more, but I'm knot really qualified to speak about the connections to other fields. So that's basically the tl;dr of why so many people care about knots!