IP Addresses are issued from a central authority, the Internet Assigned Numbers Authority. This organization handles centralized numbering for many internet resources, including registering port numbers for services and also keeping track of internet-related numbers such as ISN/ITADs for Telephony routing over IP (Current List of registered ITADs.) Various organizations across the globe, called Regional Internet Registries (RIR's) make requests of the IANA to receive IP address space that they can then allocate to ISP's and end-users.
When you or your ISP make a request for IP address space from an RIR, the RIR officially allocates the IP address space to you or your ISP, at which point the registered organization would send a BGP advertisement for that registered IP space to their
ASN. Once this has been completed, the internet has now been notified (after a brief period where BGP synchronizes) that those IP addresses are transiting to a particular location.
At this point, the IP traffic is traversing to the router where the BGP advertisements are occuring. From this point, if the operator is an ISP, they would then route you whatever IP space you requested through your existing links. If you are the operator, rather than the ISP, it would be your choice how that IP space was routed after it hit your edge router running BGP.