The adm group is for monitoring users. Zabbix is a monitoring user and should have read acces to /var/log/syslog and other stuff in /var/log* which is owned by the group adm. Whith out beeing in this group, the zabbix agent can't read from /var/log/syslog.

adm: Group adm is used for system monitoring tasks. Members of this group can read many log files in /var/log, and can use xconsole. Historically, /var/log was /usr/adm (and later /var/adm), thus the name of the group.
This if from: https://wiki.debian.org/SystemGroups