I would recommend you to look at Jones' survey paper from 1986, entitled *A New Knot Polynomial and Von Neumann Algebras.* It is very readable. Let me try to make a brief summary, though.

The basic object you start with is a $II_1$ factor. This is a von Neumann algebra $M$ with trivial center $Z(M)\simeq \mathbb{C}$, possessing a faithful trace $\tau : M \to \mathbb{C}$ (i.e. a positive normal state such that $\tau(a^*a)=0$ implies $a=0$) and having no minimal projections (this excludes matrix algebras).

For whatever reason, it is a good idea to study subfactors of $N \subseteq M$. It turns out that the most important thing about a subfactor is not so much its isomorphism type (it is very hard to tell if von Neumann algebras are isomorphic or not) but rather the way it sits inside the big factor.

Using the trace $\tau$, you can perform the GNS construction for $M$. This means that you define an inner product on $M$ via $\langle x,y \rangle = \tau(y^*x)$ (which is positive definite since $\tau$ is faithful), and then take the completion to get a Hilbert space called $L^2(M,\tau)$. Then $M$ acts on this Hilbert space by left-multiplication.

Now you do what is called Jones' basic construction. Inside the Hilbert space $L^2(M,\tau)$ is the subspace $L^2(N,\tau)$, so there is the orthogonal projection of $L^2(M,\tau)$ onto $L^2(N,\tau)$. Call that projection $e_1$. Then define a new algebra
$M_1$ to be the von Neumann algebra generated by $M$ and $e_1$ (inside $\mathcal{L}(L^2(M,\tau))$). It is immediate that $e_1$ commutes with $N$.

It turns out that if the inclusion $N \subseteq M$ was of finite index (which I won't get into here) then $M_1$ is also a $II_1$ factor (so it comes with a faithful trace also), and the inclusion $M \subseteq M_1$ has the same index as $N \subseteq M$.

Then you just keep going! Repeat the basic construction for $M \subseteq M_1$ to get a projection $e_2$, then let $M_2$ be the von Neumann algebra generated by $M_1$ and $e_2$, etc. So you end up with a sequence of projections $e_1,e_2,\dots$ which satisfy the following relations:

- $e_i e_j = e_j e_i$ if $|i-j| > 1$.
- $e_i e_j e_i = \lambda e_i$ whenever $|i-j| = 1$, where $\lambda$ is the inverse of the index of $N$ in $M$.

The projections $e_i$ give a representation of something called the Temperley-Lieb algebra.
You can see that these relations are reminiscent of the relations in the braid group. That is where the connection comes in. Knots are connected to braids, braids are connected to the Temperley-Lieb algebra (and hence to these projections) and then you can use the trace in the von Neumann algebra to define invariants of knots.

That is the gist of it. Read Jones' paper for more details.