In the words of Henry Jones Sr. "I wrote it down so I wouldn't have to remember."
Rote memory is fallible, fleeting and failing (in that order). Memorizing epic poetry is a noble cause, but even Plato would prefer to use those mental resources for applications more pertinent to a greater understanding of our physical reality. This sentiment accounts for the rapid transition to physically-recorded data, which everyone seems to agree was a general step in the right direction.
Here's a really great SF story about the effects of technology on memory (ancient and modern):
http://subterraneanpress.com/magazine/fall_2013/the_truth_of...
With writing, people did not develop memories as powerful as they did when traditions were oral -- there was no need. How many people can recite epic poems or plays from memory, let alone the Odyssey? Extremely few could these days, relatively few could 1000 years ago, and many more could 2000+ years ago. And you can bet that those who developed powerful memories in the time of Socrates would be among the most wise and cultured, so his complaint is not an unreasonable one to make. Besides, Socrates was perennially the devil's advocate and its not impossible if you had caught him on a different day he would've argued the opposite case.
Taking it further, his quote applies equally to the advent of the internet, especially search engines. Many people outsource their memory to Google -- they will even look up things on their phone in the middle of a conversation because they've forgotten its name.