Where Does Hydrogen Sulfide Come From?

The good news is that Saccharomyces cerevisiae ferments sugar to alcohol and in the process produces wine. The bad news is that S. cerevisiae is the culprit that produces hydrogen sulfide (H₂S), the odor of rotten eggs. While alcohol in wine is appreciated, wine smelling of rotten eggs is not. Moreover, the sensory threshold for hydrogen sulfide is on the order of 1 μg/l or 1 part per billion (1 ppb), a very small amount. Yeasts can use elemental sulfur, sulfate, sulfide, sulfite, and organic sources of sulfur in grape juice to produce hydrogen sulfide. ¹

The yeasts have to produce hydrogen sulfide because this is an intermediate in the biosynthesis of sulfur containing compounds required by the yeasts for cell growth and function. The two times when yeasts can create a surfeit of hydrogen sulfide are (1) during the exponential growth phase and (2) at the end of fermentation when autolysis of yeasts can lead to hydrogen sulfide production, especially if left on the lees too long. But, it's not fair to put all the blame of hydrogen sulfide production on the yeasts. Other contributing factors include:

The above list is by no means a comprehensive one, but it is a start at untangling a multifactorial problem which can result in excessive hydrogen sulfide production.
References:
1. Bruce Zoecklin, Factors Impacting Sulfur-Like Odors in Wine and Winery Operations, 8th Annual Enology and Viticulture British Columbia Wine Grape Council Conference, July 23-24, 2007.
2. UC Davis OnLine Course, VID254 Wine Quality
3. R. S. Jackson, “Wine Science Principles and Applications”, Third Edition, 2008, Academic Press.
4. Seung K. Park, Roger B. Boulton, and Ann C. Noble, “Formation of Hydrogen Sulfide and Glutathione During Fermentation of White Grape Musts”, Am. J. Enol. Vitic., Jun 2000; 51: 91 - 97.