This is on page $121$ of Hatcher's Algebraic Topology.
$Y$ is a convex subset in some Euclidean space. The linear maps $\Delta^n \to Y$ generate the subgroup of linear $n$-chains on $Y$, $LC_n(Y) \le C_n(Y)$.
For each $b \in Y$, a homomorphism is defined by taking a linear $n$-chain on $Y$ to a linear $(n+1)$-chain on $Y$, i.e., $b:LC_n(Y) \to LC_{n+1}(Y)$, via $[w_0\cdots w_n] \mapsto [bw_0\cdots w_n]$.
However, what if $b$ is a vertex in an $n$-simplex? We have $b([bw_1\cdots w_n])=[bbw_1\cdots w_n]$.
Is this well-defined? What is $[bbw_1\cdots w_n]$?
Hatcher also writes:
Geometrically, the homomorphism $b$ can be regarded as a cone operator, sending a linear chain to the cone having the linear chain as the base of the cone and the point b as the tip of the cone.
A linear map $\Delta^n \to Y$ doesn't have to be injective. If $b$ is already contained in an elementary chain $[w_0,\dots, w_n]$ then $b([w_0,\dots, w_n]) = [b, w_0, \dots , w_n]$ is a degenerate simplex. This is allowed in the definition of singular chains as well, for example for every $n$ the constant singular chain $\Delta^n \to *$ is a valid element of $C_n(Y)$.
For the "geometrically" part, if we have a $d$-chain $c = \sum_{i=0}^k a_i\sigma_i$ where $a_i \in \mathbb{Z}$ and $\sigma_i = [\sigma_{i,0},\dots, \sigma_{i, d}]$ is an elementary linear simplex then $$ b(c) = \sum_{i=0}^ka_i[b, \sigma_{i,0},\dots, \sigma_{i, d}]$$ or in other words, for each elementary simplex in $c$ we are taking the cone with $b$ as the cone point (the operation of "taking the cone" might leave the image of the simplex preserved, as in the first paragraph), and then we extend linearly to $C_n(Y)$.