A mesh network establishes a wireless interconnection among the access points. Performance is better with a wired or other out of band connection, but of course that has higher installation costs and is not practical in many situations.
The interlink of access points is so that users of any AP can reach the Internet. It doesn't have anything to do with how the user devices connect to their access point. Ways to steer users to the best AP for them is an entirely different area of study.
BATMANadv is a routing system which automatically figures out which nodes can link to each other directly and which need to relay traffic. It is an under layer to a radio system. Previously it was common to use adhoc mode as the radio layer. The 802.11s standard includes both a radio layer and a routing layer. The routing in 802.11s is very basic but is fine for a network of only a few nodes.
Lately most commercial and free projects have moved to using only the radio layer of 802.11s with the 802.11s routing disabled, then linking this radio to BATMAN. BATMAN does several things that internal 802.11s routing cannot, for example it can encapsulate VLAN tags in the packets.