Disabling 3rd party cookies is largely a privacy measure.
Cookies are restricted with a Same Origin Policy (SOP), which essentially mandates that a site on domain A can't set a cookie for domain B, and your browser shouldn't send cookies for domain A when visiting domain B. This is largely for security reasons: you don't want sites being able to read session cookies from other domains!
Advertisers and analytics companies have, of course, found ways around this. Instead of setting a cookie for their domain, they include a small resource (usually called a "pixel" tracker) on the page, from a 3rd party domain. The HTTP response from the 3rd party domain is free to set cookies for its own domain, so it generates a unique ID for your visit and sends it as a cookie. You then visit another site, which also uses that same tracker pixel. The same 3rd party domain can read back that cookie because it's part of the same origin, and therefore track that you came from one site to another.
Blocking third party cookies stops this behaviour. It prevents the 3rd party resource from setting a cookie because it was fetched in the context of a different domain, i.e. the page you're viewing. The implementation essentially modifies the cookie SOP to limit the cookie origin to the domain of the page you're viewing.
Keep in mind that this isn't infallible, though. One work-around that is commonly used is to generate a unique resource (e.g. a dynamically generated CSS file with a unique ID inside it) and serve it with cache-control headers set to ensure caching. Modern browsers hash the contents of the cached resource and send that hash as part of a HEAD request to sites to see if they've changed and need to be re-downloaded. The tracking server then takes that hash and looks it up against its database of unique IDs, and works out which one you were. Evil, no?