The answer is time. The emulsion on the paper continues to be sensitive to environmental conditions such as exposure to direct light, temperature, and humidity. Back in the 60's, photographic papers did not have the same archival lifespan as some of the newer papers of today. Depending on the type of paper, the color shifts can be different, but are usually caused by temperature and exposure to light. A film processing technique known as xpro (cross processing) can also produce similar color shifts before the printing process, achieved by processing a color slide film (reversal) in color negative chemistry (C-41), usually, but also vise versa with varying success (I have usually only crossed slides using C-41 -- doing C-41 negs in reversal chemistries usually result in very low contrast, low saturated and dull images due to the orange mask of the C-41 negs) - different film emulsions and exposure techniques result in different color shifts, and can be quite random, but is a way to achieve the "look" without software post processing in a program like photoshop. This is a property of the film, and/or paper emulsion properties, NOT the camera used. I want to make that perfectly clear. There are some who will likely note at some point and shoot camera, in-camera processing that may produce similar effect, but is not considered the same process, and will likely not produce the same effect. If you are not familiar with film and processing techniques, I would either learn how (involving a strong knowledge of film emulsion properties, and capable point and shoot or SLR film camera), or resort to digital capture and software post process technique such as, curve channel mixing or color fill layering and high pass filtering to attain the desired effect -- which also requires strong knowledge in such software.
For a DIY "vintage" look:
Here's a project for you -- Find yourself any camera that can shoot film, and old fuji point and shoot, or nikon N90 film SLR (which can be had cheap these days -- and recommended). Shoot a roll of fujichrome Superia 200 color slide film, with a one stop push (rate the ISO as 400, manually), for a stronger yellow/green color shift, box speed for less pronounced shift and muted contrast. Process in one of two ways: Find a pro lab in your area with a 3 hour lead processing time for one day service (which is common for small processing requests). Pro labs can produce the best results with a more controlled process, and can also be more expensive. Or, you can simply take the roll to target and ask them to run it last, before they change out their chemicals for the day, then run the prints without corrections. Sometimes, if you get someone familiar with film (in general), you'll find they are happy to do it for you. Ask them to run it last because the cross process can alter the chemistry, effecting subsequent batches (they usually get in trouble for doing this, so if they run it last, before they change the chemicals out, it's a win-win as the tech doesn't get into trouble for screwing up peoples film after yours, and you get your xpro).
Believe it or not, many people like the color shifts of the "vintage" or xpro look in film and film prints. It's a technique now many decades old by now and is nothing new. If you want the look, try this technique out and see how it works out for you. Either process you decide on will require some work on your part - there is no "easy" photographic technique. So don't get discouraged.