Some of the stuff on that table makes sense if you think about it, others you just need to go with, but I'll try the logical stuff first:
Overall, Pre-renal is the kidney sensing low volume. Renal is a filtration problem. And post renal is a mechanical obstruction problem.
Urine Osmolarity: Pre-renal will have a high osmolarity/cocacola coloured urine because very little is being filtered because there's a blockage before the kidney. If only small volumes of blood are being filtered by the kidney it will sense a low pressure and retain as much water as possible, making the urine hypertonic and coca cola coloured.
Renal and post renal ARF don't sense low volumes and therefore they have normal osmolarities.
Urine Na: Same concept, if there's low volume, Na will be reabsorbed to increase volume.
BUN/creat: BUN backs up faster than creat when there is little or no filtration. Prerenal will be greater than 20. Renal there's aberrant filtration so the BUN is more freely filtered than creat, so your ration is less than 15. In an obstruction, there's no problem with filtration so it's within the "normal" range of 15 to 20.
that's not everything on that table, but it's the stuff I can make sense of. Hope this helps!
Here's to short term memory!!!!!!!!!!!!!!!!!!!!!!!!!!