LED brightness is controlled by current, not voltage. If you're controlling brightness with voltage, there is something that is controlling the LED current based on that voltage.
Once the LED reaches forward conduction mode (when the supply voltage is greater then the forward voltage of the LED), the current needs to be limited, else the LED will pop like a fuse. That's all the Christmas bulbs usually use for current limiting; there are 30-whatever LEDs in series (the LED string itself performs the rectifier function) and a single resistor for the entire string to set the string current. That's a little reckless for a robust LED design (I would hope there is a reverse polarity diode as part of the LED string so the fragile LEDs are less stressed), but works for cheap consumer products with a very fixed supply voltage.
Because a simple resistor controls current based on the voltage across it, a variable voltage can also be used to do brightness control if things like supply voltage are simple (120VAC is as simple as it gets). Change the input voltage to the resistor, and the current through the resistor changes.
There are many LED drivers that do much more sophisticated things like constant current control, PWM brightness control, have switching power supply functions, etc., depending on what source your power is coming from and what you need your efficiency to be. Almost anything with any level of sophistication has some type of current sink driver or current controller to solve power problems or add product functionality.
LED replacement bulbs for your house generally have power supply and/or driver circuits built in, especially if they use only a high flux LEDs. With a string of only a few LEDs, you have a lot lower forward voltage than the 120VAC supply voltage, so you would generally use something like a buck converter or PFC controller to get into a proper range of DC voltage and current without wasting that big voltage delta as heat.