Here is a motivation for hocolims, for concreteness look at pushouts of topological spaces:
The usual pushout functor $colim: Top^P \rightarrow Top$, where $P:=\bullet \leftarrow \bullet \rightarrow \bullet$ is the diagram of a pushout datum, does not respect weak equivalences: Take for example the diagrams $pt \leftarrow S^1 \rightarrow pt$ and $D^2 \leftarrow S^1 \rightarrow D^2$ in $Top$. Since $D^2$ (the 2-dimensional disk) is contractible, there is a map between the two diagrams consisting of weak equivalences, i.e. a weak equivalence in $Top^D$. But if we apply colim to both diagrams we get non-equivalent objects in $Top$, namely $pt$ in the first case and $S^2$ in the second.
Thus the pushout functor $colim: Top^P \rightarrow Top$ does not take weak equivalences to weak equivalences and we can not invoke the universal property of $Ho(Top^P)$ to get a functor $Ho(colim):Ho(Top^P) \rightarrow Ho(Top)$ making the square commute which has $colim$ on top, $Ho(colim)$ at the bottom and the projections to the homotopy categories at the left and right. The best we can do is to find the initial functor "$hocolim$" which allows filling this square with a natural transformation - which is exactly the Kan extension property you cited.
For model categories (but not only there, e.g. also Baues cofibration categories) you can define that functor via the top level, i.e. going $Top^D \rightarrow Top$, in a particularly neat way. Back to the example: Topologists noticed that the colim functor, when restricted to pushout data $A \leftarrow B \rightarrow C$ in $Top$ with $B$ cofibrant and the arrows cofibrations, does preserve weak equivalences, e.g. building the pushout of two inclusions of $S^1$ into contractible spaces such that there is "enough space" around the image of $S^1$ inside those spaces, you will always get something equivalent to $S^2$ - try it out! So the recipe for computing the $hocolim$ is first replacing your diagram by one with these properties (cofibrant replacement - this is an endofunctor which preserves weak equivalences) and then apply $colim$ - this does now preserve weak equivalences and thus descends to a functor between the homotopy categories.
So the intuition about $hocolim$ - which is good in great generality - is that is the best approximation to $colim$ which preserves weak equivalences ("is homotopy invariant"); the cofibrant replacement construction stems from the fact that this is a class of objects where $colim$ is already homotopy invariant.
The story for homotopy limits is of course dual, instructive examples are homotopy fibers and homotopy fixed point objects.