Isn't the greenhouse effect what makes Earth livable in the first place?

Yes, the atmosphere is the covering around the Earth that holds in heat when it is not sunny. The environmental problem of the Earth overheating  is an enhanced greenhouse effect.

How does a real greenhouse work? A traditional green house is a clear glass enclosure for plants that require a warmer environment than the outside air. Solar energy comes in through the glass as light (radiation) and is absorbed by the air and materials in the greenhouse. The energy is re-emitted as heat energy, which does not pass through the glass as easily as the light energy came in. As a result, the air inside the greenhouse warms up and provides a warm environment for the plants. 

How does that concept relate to the earth? The term "greenhouse effect" is a metaphor used to describe the excess warming of the earth due to the addition of "greenhouse gasses".  The earth's atmosphere acts like the glass in a greenhouse, allowing sunlight in to be absorbed by the earth and converted to heat energy. Without the atmosphere and its greenhouse gasses, the earth would not hold heat and would not be able to support life as we know it. The additional carbon dioxide and other greenhouse gasses in the atmosphere are causing more heat to be absorbed in the atmosphere and on earth, and temperatures on earth to rise. This is sometimes called the "enhanced greenhouse effect".