Do contemporary publications cite more sources? Why? When I read very old, classic papers in biology, I am struck by how few sources they cite. Watson and Crick, 1953 cites 6 sources. Luria and Delbrück, 1943 cites 9. Sanger, 1977 cites 14.

This is in contrast to contemporary papers, which often have a whole page, in very small type, listing the sources.

What is going on? Am I correct in concluding that over the past century, the number of sources referenced by each paper has increased? What does this mean? Is biology (and perhaps other sciences) simply maturing and becoming more collaborative? Has the availability of computers and the internet made it easier to find more sources? Are standards higher nowadays about grounding your work in literature? Is it just that stuffing your bibliography with copy/pasted sources has become fashionable?
jakebeal has pointed at the more technical reasons for the increase of references. We should, however, not forget some of the more subtle, but perhaps also, more fundamental changes that has happened.

First, there is a gradual change in how science is communicated. Scientific papers have developed from letters that were read aloud in front of scientific societies and also published as personal letters. The scientific debate was more closely akin to debates between persons making observations. Obviously this was possible for the reason that there were very few involved in any one research question. So, in part the development seen is due to a development of publishing driven by changes in the form and volume of debate. 

Second, research questions have become more complicated involving larger and larger groups of scientists with varying expertise. This increase in complexity also means references are no longer required to cover just a specific question but also information from adjacent or supporting fields.

Third, science is disseminated in smaller parts today that what was the case back in time. This is partly out of necessity related to the second point above. Additionally, and this is perhaps not the greatest aspect of developments, there is the pressure to publish due, mostly, to the fact that academic careers are measured in terms of number of publications and number of citations. The number of publications has thus increased for several reasons and hence also the number of somewhat relevant papers to cite. I will not get into bad behaviour such as self-citations here but it is clear that any system will have flaws and some people will make use of such flaws to benefit themselves.

So at least some of the change in number of publications is due to developments in the way we perform and communicate science and also changing pressure from the academic world on researchers to publish and be cited.