The question is as the title asks. My background is in computer science, and recently I'm trying to read a paper that involves using extremal length to prove certain properties of planar graphs.
I have a little bit of difficulty understanding the problem conceptually, as far as I understand it's a measure of a upper bound to cross some shape, there is a few examples with rectangles and annuli on wikipedia, but it's a bit hard to understand what its doing and what its for.
Intuition
If we think of curves as roads, then the extremal length of a family of curves describes how long it takes for a lot of people to drive along them from end to end, accounting for traffic in congested areas (where many roads merge). If set $A$ is a suburb and $B$ is a business area of a city, then the extremal length of all roads connecting $A$ to $B$ describes how long the average commute time is.
Invariance
The main property of extremal length is its conformal invariance: if $\Gamma$ is some family of curves, and $f$ is a conformal map of a domain that contains all curves in $\Gamma$, then the family of images $\{f\circ \gamma : \gamma\in\Gamma\}$ has the same conformal length as $\Gamma$ itself.
Moreover, this invariance is robust in the sense that if $f$ is only quasiconformal (has a controlled deviation from conformality), then the extremal length of the image is distorted only by a certain multiplicative amount. One might say it's quasi-invariant under quasiconformal maps.
So, on one hand the extremal length is a geometric concept; on another, it is related to the analytic properties of maps such as Cauchy-Riemann or Beltrami equations. This makes it a useful tool for deriving geometric conclusions from analytic assumptions, or vice versa.
Reference
The classical book Lectures on Quasiconformal Mappings by Ahlfors has both an accessible introduction to extremal length, and its applications such as a sharp Hölder continuity estimate for quasiconformal maps of the unit disk.