The argument in favor of waiting goes something like this: By holding out until your full retirement age, if not longer, you increase the chances of maximizing your cumulative lifetime benefits. This follows from the fact that the size of your monthly checks is determined in part by when you apply for Social Security. If you apply at age 62, which is the earliest allowed, then your benefits will be approximately 25% lower than your primary insurance amount -- that is, the benefit amount you're entitled to upon reaching full retirement.
By contrast, if you wait until you turn 70, which is the latest age at which you can claim benefits, then your benefits can be as much as 32% larger than they would have been had you taken them at age 66. In official Social Security jargon, these are known as "actuarial adjustments."
Seems pretty cut-and-dried, right?
Well, it turns out that it's not as straightforward as it might first seem. This is because people who elect to take benefits early get more checks than those who wait. It's a trade-off. Do you want more checks that are smaller and come sooner? Or would you prefer fewer checks that are larger but come later in life?