A singular $n$-simplex $\alpha: \Delta^n\rightarrow \Delta^p$ in $\Delta^p$ is called affine if $\alpha(\sum_{i=0}^n t_i e_i)= \sum_{i=0}^n t_i\alpha(e_i)$ holds for all $t_i$ with $\sum_{i=0}^n t_i=1$ and $t_i\geq 0$. Following Hatcher, given an affixe $n$-simplex $\alpha$, we set $v_i=\alpha(e_i)$ and write $[v_0,\ldots, v_n]$ for $\alpha$.
Let $K$ be a set of affine simplices in some fixed $\Delta^p$. In my definition of a simplicial complex $K$ we speak of the intersection $\sigma\cap\tau$ of two affine simplices $\sigma\cap \tau$, and require $\sigma\cap \tau\in K$.
What does $\sigma\cap\tau$ mean? What I can think of is that we intersect the convex sets $\operatorname{im}(\sigma)$ and $\operatorname{im}(\tau)$ and then require that there is a affine simplex in $K$ which has an image equal to this intersection. Is that correct? Implicitely we are identifying maps with their image.
I am surprised that your lecture notes do not contain a proper definition of an affine simplex. An affine (or geometric) $n$-simplex in a Euclidean space $\mathbb R^p$ is the convex hull $[v_0,\ldots,v_n ]$ of $n+1$ points ("vertices") $v_0,\ldots, v_n$ in general position (this means that the vectors $d_i = v_i - v_0$, $i = 1,\ldots, n$, are linearly independent). If you want, you can replace the ambient $\mathbb R^p$ by some fixed affine simplex $p$-simplex $\Delta^p$.
The requirement $\sigma \cap \tau \in K$ (more precisely, $\sigma \cap \tau = \emptyset$ or $\sigma \cap \tau \in K$) means that $\sigma \cap \tau$ has to be the convex hull of a set of vertices which are common vertices of both $\sigma$ and $\tau$.
See for example Chapter 1 of
Also see my answer to What is the difference between cellular, simplicial and singular homology and their simplices?