The following definition is taken from Daniel Murfet's Triangulated Categories Part I notes.
Let $\mathcal T$ be a triangulated category with countable coproducts. Suppose we are given a sequence of objects and arrows in $\mathcal T$ $$X_0\overset{j_1}{\longrightarrow}X_1\overset{j_2}{\longrightarrow}X_2\overset{j_3}{\longrightarrow}X_3\overset{j_4}{\longrightarrow}\cdots$$ Let $\mu:\bigoplus _{i=0}^\infty \longrightarrow \bigoplus _{i=0}^\infty X_i$ be the arrow induced out of the first coproduct by the arrows $j_{i+1}:X_i\rightarrow X_{i+1}$. That is, $\mu u_i=u_{i+1}j_{i+1}$ where $u_i$ is the injection of $X_i$ into the coproduct. A homotopy colimit of the sequence, denoted $\underrightarrow{\mathrm{holim}}X_i$, is a homotopy cokernel of $1-\mu$. That is, it an arrow $v:\bigoplus_{i=0}^\infty X_i\rightarrow \underrightarrow{\mathrm{holim}}X_i$ fitting into a distinguished triangle $$\bigoplus_{i=0}^\infty X_i \overset{1-\mu}{\longrightarrow} \bigoplus_{i=0}^\infty X_i \overset{v}{\longrightarrow} \underrightarrow{\mathrm{holim}} \overset{w}{\longrightarrow} \Sigma \left\{ \bigoplus_{i=0}^\infty X_i \right\}$$
I'm looking for intuition for this definition, and also for proof that it coincides with the general definition as a derived functor of the usual colimit functor. References are welcome.