The kidney has a number of jobs; filters waste products and toxins from the blood, maintains pH of the blood, maintains hydration and produces certain hormones. When the kidneys are not working properly all these functions are affected, but the most significant [initially] is the build up of waste products in the blood.
If you appreciate that blood is made up of three broad components; red blood cells [carry oxygen], white blood cells [fight infection] and plasma [liquid bit with stuff dissolved in it and various things such as protein].
The simplest analogy of a kidney is a sieve. As blood passes through the kidney the blood cells are kept in the blood stream but most of the liquid is filtered out. In a normal kidney protein in the plasma is quite large and has a negative charge so it doesn't get filtered out. As the filtrate moves through the kidney some things are excreted and made into urine, some are reabsorbed.
When the kidney is not working think of the sieve with bigger holes. Now the protein is also leaking out. Protein in the urine is one of the very early signs of kidney disease. As the body loses protein it requires more good quality protein to replace losses. As there are less working functional units in the kidney it also means the kidney is less able to reabsorb water and you pee more. Because you are losing more water, your body gets dehydrated and you drink more - another sign of kidney problems. Because the kidney is not working efficiently phosphorus builds up in the blood, and if increased it's a sign the disease is more advanced.
Phew. So now we know what's happening we need to know how we can help. Well we need a diet that is high in good quality short chain proteins and low in phosphorus. We want a diet that produces less 'waste' for the kidneys to deal with. We also need it to be palatable (they often lose weight because [in part] of the loss of protein and the build up of waste products leads to nausea in people and we assume dogs). It is also VITAL that we encourage water intake - the more fluid passing through the kidney, the more chance it has of washing away waste products.
Depending on the practice would depend on the food suggested but the two most commonly prescribed are Eukanuba Renal or Hills K/D. They really do help to prolong life - but nothing will 'cure' the underlying condition.
Does that help?