For each elliptic curve $E$ over a number field $K$, the plot displayed consists of the graphs of the real points of $E$ with respect to each real embedding of $K$. Complex embeddings of $K$ are ignored, so for example there is no plot shown when $K$ is a totally complex field such as an imaginary quadratic field.
The plots are produced using SageMath, which uses plotting logic designed by David Roe to determine which subset of $E(\mathbb{R})$ in the real affine plane to include in the plot.