The other answers hit on some good points, but they are not all 100% correct. If they advertise fiber to the home then they must actually be running fiber to your property. They can convert it to copper at that point, but if they don't get the fiber to your property then they can't be calling it "Fiber to the Home".
What originally caused phone lines to be slow for data communications was the low pass filter that was placed on the phone line. zebonaut is correct in his definition of channel capacity. It has been awhile since I have messed with that stuff, but I believe the low pass filter was set around 8 KHz.
DSL then came around and took advantage of the fact that the phone companies had physical access to the lines before the low pass filter. This meant that they could use frequencies that voice traffic didn't exist in but that the copper could transmit. There was still low pass filters put in place for voice traffic so that you wouldn't here high pitched noise on your phone call or have any aliasing problems.
DSL was limited in speeds based off of the S/N portion of zebonaut's channel capacity formula. The longer the wire was the worse SNR you would get. As time as gone on phone companies have continually shortened the copper loop and replaced old copper with fiber. As this has happened DSL speeds have been able to increase.
It has gotten to the extent in many areas that there is fiber almost all of the way to the house. This as made it so that the cost to get fiber all of the way the property is actually rather reasonable. This also provides the ability to advertise "Fiber to the Home" and like I said earlier they can't call it that unless it actually is that.
One example of a situation of not having "Fiber to the Home" is AT&T's U-verse. In many cases they actually do have fiber to the home, but because they don't have that everywhere they don't advertise "Fiber to the Home", but rather they advertise "fiber optic technology and computer networking".
Fiber versus Copper
One of the big reasons we use fiber instead of copper is how much noise copper can pick up where as fiber picks up almost no noise. This makes the SNR very good which allows for much higher data rates.
By getting fiber all of the way to the property, the length of copper needed is minimized as well as there are no neighboring copper lines that can cross talk into your line.
So basically, the less copper you have the higher the data rate you can get.