That is because you have not enable API in config file. Go you your testlink installation directory and edit config.inc.php. Change the following setting to TRUE. Next, restart Apache or whatever web server you use. Login again and visit "My Settings" page (link on top of each page). You should see a section labeled "API Interface" and a button "Generate a new key".
/** XML-RPC API availability (disabled by default) */
$tlCfg->api->enabled = TRUE;