BGP synchronization was an old routing loop prevention mechanism that worked in the following way:
A BGP border router would not propagate a BGP advertisement to an eBGP peer until the router does see the same IP prefix learned in IGP ( OSPF, ISIS or other internal protocol).
This would ensure that there are no black holes inside the ISP AS caused by routers not talking BGP. Intermediate routers only taking part in IGP would know how to route traffic with this destination.
In old times not all routers were BGP speakers so the rule was added by Cisco and enabled by default.
The rule is now by default disabled.
The rule can be disabled when:
– all routers in the AS take part in iBGP ( using BGP RR or confederations this is easy to achieve)
– in modern MPLS based networks internal routers ( P nodes) do not perform IP routing but MPLS switching so they can be unaware of IP prefixes and can be not part of the iBGP mesh.
So it is something that you should not worry about, it is good to know what is, but it is disabled by default.