MPLS does not go into the "Internet" MPLS has several components, but it's mainly a switching mechanism used by the routers in the ISP networks. A router running MPLS switch the packet (the IP datagram) based on a label, instead of using the L3 address. This is supposed to be more efficient since the label is shorter and the FIB can be populated more efficiently.
For the routers to know what label to use and what to do with them they use label distribution protocols or manually configured labels.
If a router does not participate in the MPLS distribution will not know what a label means, and it will discard the packet with the label, or it will not understand this is an IP packet at all. As the organization running the network controls which routers participate in the MPLS, it is considered that the traffic is safe inside the network. For building VPNs in the MPLS network what they do is stack labels, which requires an additional label distribution, normally made using BGP extensions. So, as long as the MPLS runs only in the provider network it is considered safe.