Actually, since you are eligible to take social security the first whole month in which you are 62 or any time after that, that is not strictly correct. But that's basically how it works - you get a social security check the month after you actually start, which is the first month you are 62 for the whole month or any month thereafter that you have asked them to start.
If you were starting at 62 and your birthday was 15 Feb 1963, you would get your first check on 16 Apr 2025, which is the third Wednesday after the first month of your eligibility.