What's the etiquette for following up on academic contacts? It's happened a few times to me that I met a professor or researcher at an informal event, we had a good conversation, and they offered "help," as in "if you need any help, just shoot me an e-mail" or "let's keep in touch, in case you need help in the future." I have no idea what "help" is, or how I am meant to follow up on such an offer. Is it a sort of platitude that I am not meant to take literally?

These are usually not researchers or professors at my university, but they are usually in my field. I'm an undergraduate, currently in my final year and applying to grad school, but obviously this hasn't been the case for the past few years (and this is a recurrent situation).
As far as I see, these phrases mean what they say (which is not that much, but more than nothing). I also hear them a lot and say them from time to time. However, they do not promise anything else than some interest in your work or field.

When I say these phrases to people I just met it means that I found their work interesting and can imagine to spend more time thinking about it (but probably will not think about it on my own). 

Another view is: Many people collaborate with many other people and enjoy it. See this phrase is a very low level entrance to something that may be a collaboration in the future. 

More on the practical side: You may always send an email as a follow up, either with a question or something like "I enjoyed our conversation; here are links to some of my papers/other interesting stuff on the topic. Looking forward to seeing you again."