What does the span argument control in geom_smooth?

The span (also defined alpha) will determine the width of the moving window when smoothing your data.

"In a loess fit, the alpha parameter determines the width of the sliding window. More specifically, alpha gives the proportion of observations that is to be used in each local regression. Accordingly, this parameter is specified as a value between 0 and 1. The alpha value used for the loess curve in Fig. 2 is 0.65; so, each of the local regressions used to produce that curve incorporates 65% of the total data points. "

Taken from:

Jacoby (2000) Loess:: a nonparametric, graphical tool for depicting relationships between variables. Electoral Studies 19-4. (Paywalled paper)

For more details check the referenced paper.

LOESS smoothing is a non-parametric form of regression that uses a weighted, sliding-window, average to calculate a line of best fit. Within each "window", a weighted average is calculated, and the sliding window passes along the x-axis.

One can control the size of this window with the span argument. The span element controls the alpha, which is the degree of smoothing. The smaller the span, the smaller the 'window', hence the noisier/ more jagged the line.

Look for documentation under LOESS rather than span.