Question
   How does the thickness of grass effect its ability to absorb water.

Answer
Water is taken up by the roots of grasses. Imagine a very light rain (or irrigation) falling on your lawn. At first the rain drops wet the leaves; then as the leaves get completely wet, the water starts to trickle down to the ground where it wets the grass runners, litter of decaying vegetation etc. Water next reaches the soil and only then does it become available to the grass roots. After the rain/irrigation has stopped the process will continue; in similar fashion as water continues to drain out of a potted plant after you have stopped watering it.
You can imagine that the thicker the grass, the longer it will take for the water to reach the roots.
Does this answer your question? Please come back if it doesn't.
Interestingly, this is one of the benefits of a grass cover. It traps/slows down precipitation and allows water to penetrate the soil. This prevents or delays runoff thereby reducing erosion. The overall benefit is water conservation.
I just thought I'd mention that since lawns are getting such a bad name these days.