When traffic arrives into a PE router from a customer the interface that it arrives on is associated with a VRF. The router will only lookup the destination in the VRF routing table for the associated VRF. If the interface is associated with VRF A, the router will only be able to see the 188.8.131.52 that exists within VRF A. If the interface is associated with VRF B, the router will only be able to see the 184.108.40.206 that exists within VRF B. The router will either forward the traffic locally out of another interface associated with the same VRF, or if the destination is on another PE router the traffic is encapsulated in MPLS and forwarded to the egress PE router.
If you are familiar with VLANs, VRFs are very similar, but at L3 instead of L2. When a frame arrives on switch, the port is associated with a VLAN and the switch looks up the destination MAC address in the MAC address table for that VLAN. The same concept exists for VRFs at layer 3. When a packet arrives on a router, the interface is associated with a VRF and the switch looks up the destination IP address in the routing table for that VRF and then forwards out of that interface (which is also in the same VRF).