Pennies ain’t got 'em, nor do nickels. Dimes got 'em, as do quarters. Why do some denominations have grooves but not others? I happened to have a 10 centimes coin here in my drawer - it ain’t got 'em. The 5 franc coin does. Does it have something to do with the materials they’re made of?
The U.S. Mint’s website provides a wealth of information.
Summary: The practice was introduced largely to prevent “clipping” the edges of coins made of precious metals, and keeping the metal either for sale or counterfeiting. The dime is reeded to make it more distinguishable from other small coins by touch.
Exactly so. Cheapskates used to clip off the edges of coins – this was back when they were made from precious metals – and pass them off as whole. The amount of silver that could be taken off any one coin was negligible, but clip a few and your pile is worth something. So, the milled edge was invented. In the U.S., pennies and nickels are, literally, small change, so it was not felt necessary to guard against the clipping of those denominations.
Milling is a holdover from the days when the metal in the coins was worth something. It was intended to keep people from shaving the coins and collecting the gold or silver from them. It was only done to the coins made of more valuable metal where it would be worth it to shave them. These days, it pretty much simply serves as a way of differentiating the various coins.
I heard somewhere that Sir Isaac Newton invented the process (and got his knighthood from it) but these milled coins certainly predate Newton (1643-1727). Another good story out the window. 
So who did start it?
Note that there is a distinction between “milled” and “reeded”. The coins under discussion are “reeded” coins–“milled” simply means that the coins were stamped by machine rather than hammered. According to the World Coin Identifier site, reeded edge coins are a 19[sup]th[/sup] Century invention; this may refer only to reeding on milled coins. However, according to this History of Money, reeding was demanded and (presumably) implemented in the Roman Empire. The implication in the essay is that this predated the debasement of the denarius with base metal, from which we can deduce that reeding was probably implemented no later than 54 AD.
The chances of figuring out exactly whose idea it was are slim, to say the least.
Balance I wouldn’t put too much trust in History of Money as a source of correct information concerning coin minting techniques. I found more than a few errors in a quick reading.
While I can’t dismiss their claim about the Romans inventing reeding on coins based solely on my memory, I don’t think this is true. I have handled thousands of Roman coins, copper silver and gold, and they don’t have reeds.
Edge devices or designs were first put on coins around 1555 in France with a segmented collar. In 1685, Castaing invented a machine which would also imprint the edges of the coin blanks with a design or letters, either incuse or raised but before they were struck.
The vertical reeds which we have on US coins today were first applied in 1836 using a segmented collar and a steam press.