Loop spaces have the homotopy type of a topological groups

Here is a very naive explanation with all technicalities skipped.

Let $X$ be a simplicial set (this is a type of discretisation of a CW complex which is a generalization of a simplicial complex). In his paper Kan constructs out of $X$ another simplicial set, which he denotes by $GX$. The set of $i$-simplices of $GX$ is the free non-abelian group with generators the $i$-simplices of $X$ (with a small part of them cut out, but that is not that important here). The obtained simplicial set $GX$ is thus a simplicial group, meaning that for every $i \geq 0$ the set of $i$-simplices of $GX$ is a group and in addition, the face and degeneracy operators are homomorphisms (faces are defined in a similar way as for a simplicial complex). If one is familiar with simplicial sets, it is not hard to see that the geometric realization of a simplicial group is a topological group (the geometric realization of a simplicial set is defined similarly to the geometric realization of a simplicial complex and is a CW complex).

The crucial thing in that part of Kan's paper is to prove that $GX$ is a model of a loop space for $X$. This goes as follows:

Step 1: Out of $X$ and $GX$ Kan constructs another simplicial set $GX \times_\tau X$ ($\tau$ is a certain explicit map of simplicial sets which comes naturally together with the construction of $GX$) Step 2: Prove that the projection $GX \hookrightarrow GX \times_\tau X \xrightarrow{p} X$ is a Kan fibration (the equivalent of a Serre fibration but in the category of simplicial sets ; it has identical properties) Step 3: Prove that the simplicial set $GX \times_\tau X$ is contractible (this part is rather technical)

So, the geometric realizations of $GX \times_\tau X$ and $GX$ give a path-space and loop space of $X$, thus they are homotopy equivalent to the standard path space and loop space of $X$, and moreover, the geometric realization of $GX$ is a topological group.

There are a couple of other nice places where you could read the details such as:

Peter May - Simplicial objects in algebraic topology - a more combinatorial approach Goerss, Paul G., Jardine, John - Simplicial homotopy theory - a more category theoretical approach

I hope that was useful.