Understanding need for PKI
Your understanding is correct. PKI would be the service that verifies that Bob's public key really belongs to someone named "Bob". As long as Alice trusts the certificate authority that certified Bob's public key, she can implicitly trust that the public key she sees really is Bob's, not Eve performing a MITM.

The internet is secured with web PKI, which is the network of Certificate Authorities that certify that a websites' public key really belongs to the domain it claims to belong to (for Domain Validation, or DV). It's far easier to trust a handful of CAs than it is to trust every single website you'll ever visit.