The paragraph after the charts explains what is meant:
Note: plots show turnout (number voting/number eligible) and vote proportions (numbervoting for party/number eligible) for (a) the Democratic Party or (b) the party the mostvotes in each constituency in aggregation units in the Korea 2020 parliamentary election.Plots show scatterplots with estimated bivariate densities overlaid, with histograms alongthe axes. 328 “abroadoffice” observations reported with zero eligible voters but often witha positive number of votes are omitted.
The horizontal axis show proportion of turnout (number voting/number eligible), and the vertical axis shows proportion of leading votes (number voting for party/number eligible). In other words, as I understand it, the vertical axis shows the number of pro-party voters as a proportion of the total eligible voters.
The legend description "Prop. of Leading Votes Among Eligible Voters" is not idiomatic in my mind, so your confusion is definitely understandable. I had to read the paper in question to understand what that label was expressing.