You can read in Wikipedia the following:
Given a space X and a loop $\alpha\colon S^1 \to X$ representing an element of the fundamental group of $X$, we can form the mapping cone $C_α$. The effect of this is to make the loop $α$ contractible in $C_α$, and therefore the equivalence class of $α$ in the fundamental group of $C_α$ will be simply the identity element.
So we have $[\alpha (x)]=[(x,1)]$. Does it mean, that $\alpha$ is homotopic to the constant path in $[(x,0)]$ by just moving the loop continuously to the peak of the cone? Or why is the equivalence class of $\alpha$ the identity element?