A vulnerability exploitable with only local access requires the attacker to have either physical access to the vulnerable system or a local (shell) account. Examples of locally exploitable vulnerabilities are peripheral attacks such as Firewire/USB DMA attacks, and local privilege escalations (e.g., sudo).
Quote from the CVSS specification
Vulnerabilities with this tag were given a CVSS rating as part of the requirement to be included into the National Vulnerability Database. You can learn more about what the individual scores mean in the CVSS specification document.