Because everybody dies.
That's always been true. But death now is a bit different from what it was a thousand or even a hundred years ago. We no longer die in childhood, or in childbirth. We no longer die of epidemic disease, starvation, squalor and poverty. Very few of us die in accidents - only 1,257 of 29,204 deaths in 2009. Deaths due to crime are insignificant, and war even more so. If you are reading this you can expect to live to a ripe old age. And if you don't pop your clogs quickly due to heart attack or stroke, then you can expect to die slowly in a hospital bed, probably of cancer.
This is a Good Thing, and something we should thank generations of doctors, scientists, sanitation engineers and policymakers for. But the flipside is that an increasing proportion of us are dying long and lingering deaths. Thanks to the miracle of modern medicineTM that isn't necessarily painful - but pain management is imperfect and there are no drugs for dignity. It is distressingly common for this process to result in people wanting to die early, while they still have some control and dignity left to them. Not so common that you can expect it to happen to you (for a start, see heart disease and stroke), but common enough that it will probably happen to someone you know, possibly to someone you actually care about.
At the moment, people solve this themselves, and we call it murder or suicide depending on the exact method and who is involved. It means people being prosecuted for helping their loved ones in their last moments, and it means people choosing a slow and painful death by starvation to avoid imposing on their friends. You don't need to have seen this first-hand to recognise the cruelty and absurdity of it. We have people who want to die, and perverse rules which will not let them.
And that's why we need a death with dignity law: so that people aren't prosecuted for an act of kindness. So that people aren't forced to choose a lingering means of ending their lives. So that people unlucky enough to die in this way have a choice about the matter, rather than being cruelly forced by law to suffer.