Maybe Sojourner's response already made it clear enough, but:
Suppose I only told you I was traveling at 3meters per second. You could know the total distance I'll travel in a given number of seconds by multiplying that number of seconds by 3 (6 meters in 2 seconds; in 5 seconds, 15 meters, and so on...). So rate of speed is a matter of multiplication, and the conjunction "per" actually means "multiplied by", or "times", and "meters per second" actually means "meters x seconds".
Well, I haven't given you enough information to tell what my rate of acceleration is, have I? But, if I tell you that I was standing still a second ago, and I'm moving at three meters per second (ms) now, and then a second later I tell you that I'm going six ms, you can determine a pattern. Although I've only reported my speed (in ms), you can see that every second, I'm going three meters per second faster than I was the second before, so I'm gaining three ms each second (or "per second"). So, you can tell how fast I'll be going a given number of seconds from now by multiplying that number (of seconds) times the 3 ms I'm gaining each second (after 1 second, I was going 3ms, after 3 seconds, 9ms; at 10 seconds, I'll be going 30ms and so forth...). Hence, three "meters per second", per second, or (since "per" means "times") "meters x seconds x seconds". And, since to multiply any number times itself is to square that number, "seconds x seconds" = "seconds2", so 3 mss = 3 ms2.