A quantum point contact (QPC), a narrow region separating two wider electron reservoirs, is the standard building block of sub-micron devices. As a function of its width, the conductance through a QPC changes in integer steps of G0=2e^2/h signaling the quantization of its transverse modes. Such measurements also reveal an additional shoulder at a value around 0.7G0, an observation which remains a puzzle even after a couple of decades. The phenomenology of this "0.7 anomaly" in quantum point contacts is fully explained in terms of a quasi-localized state, which forms as the point contact opens up. Detailed numerical calculations within spin-density functional theory indeed confirm the emergence of such a state. Quantitative calculations of the conductance and the noise are obtained using a model based on these observations, and are in excellent agreement with existing experimental observations.