One reason people care about knots in low-dimensional topology is that every compact ("finite-volume") 3-dimensional manifold without boundary can be constructed by taking a 3-sphere (the set of points in R^4 unit distance from the origin), boring out tubes along a collection of disjoint knots, then gluing solid tori ("donuts") back in in a different way, a process called Dehn surgery. This is the Lickorish-Wallace theorem. Sort of the intuition is that if you take a 3-manifold and have worms eat out enough closed loops, the manifold loses its integrity and becomes indistinguishable from the complement of a collection of disjoint knots. (Lickorish's version of the proof involves a theorem that's colloquially known as the Lickorish Twist Theorem.)
In particular, every 3-manifold is the boundary of a 4-manifold obtained in a way that's reminiscent of knot traces from the article. You take a disjoint collection of knots in the boundary of a 4-ball, then glue in the 1-handles ("caps" in the article) along these knots, but with slight change: you glue in 1-handles with any framing whatsoever, not just the 0-framing like in knot traces (and actually using just +1-framing and -1-framing is sufficient). It's actually a remarkable fact in its own right that every 3-manifold bounds a 4-manifold; this is saying the 3-dimensional cobordism group is trivial. Other-dimensional cobordism groups are not trivial in general.
Every 3-manifold has a diagram, then, consisting of a multi-component knot (known as a link) with each component labeled by an integer (or a rational number if you are ok with "fake" surgeries). There is a whole thing called the Kirby calculus that gives a sufficient set of moves to go between any two such representations of a particular 3-manifold. An extension to this calculus went into Piccirillo's calculations with knot traces -- she cites the classic Gompf and Stipsicz for details.
One use of this representation of a 3-manifold is to construct Reshetikhin-Turaev invariants, which are sequences of numbers associated to a 3-manifold. This is related to the Jones polynomial, and these invariants satisfy a number of wonderful properties that together mean they form a topological quantum field theory (TQFT). I don't know the physics, but I'm under the impression you can interpret it as having something to do with quantum states of anyonic particles.
For books, you might look at Adams "The Knot Book" or Prasolov "Intuitive Topology" to get a substantial taste of knots and low-dimensional topology.