The gloves are iGlove Touchscreen Gloves, on sale for $12, including shipping. It's unclear how long the deal will last, but it does say "limited time." The gloves come in black, red, pink, light blue or grey and are hand washable.
The explanation for why you can't wear regular gloves and use an iPhone or an Android phone comes via the technology used in the touchscreens. The latest generation of mobile devices, such as Android smartphones and tablets and iDevices such as the iPhone, iPad and iPad mini all use capacitive touchscreen. These detect a person's touch by using an electromagnetic field. Since the human body is an electrical conductor, touching the surface of the screen results in a distortion of the screen's electrostatic field, which is then processed to determine the type of input.
Previously, resistive screens were used in handhelds such as Palm devices. These depend on pressure to determine when the screen is touched, which is why swiping gestures don't work on them. Two thin, transparent electrically-resistive layers are separated by a thin space, and when they touch via pressure, viola: the touch is noted. This is dated technology in consumer focused devices, but still has applications in restaurants, factories and hospitals due to its high resistance to liquids and contaminants.
It is also possible to use a resistive touchscreen with standard gloves.
Wikipedia says the following about capacitive touchscreens:
A capacitive touchscreen panel consists of an insulator such as glass, coated with a transparent conductor such as indium tin oxide (ITO). As the human body is also an electrical conductor, touching the surface of the screen results in a distortion of the screen's electrostatic field, measurable as a change in capacitance. Different technologies may be used to determine the location of the touch. The location is then sent to the controller for processing.
These gloves, and others like them, work because the fingertips are made or coated with conducting materials. Spreebird says:
These super-soft gloves have a special conductive material interwoven into fingertips which allows you to use most touchscreen devices including your iPhone and iPad with the gloves on. The one-size-fits all gloves will not scratch or damage your devices' screen and can be hand washed to keep them fresh all winter!