The main purpose of a CA is to confirm the identity of the owner of the certificate.
In case of a website, certificate confirms that the public key really belongs to this website. Otherwise any website could claim it is google.com or microsoft.com. When you establish a TLS connection to a website my-bank.com, you can be sure it is really my-bank.com and not some fishing website. Otherwise your browser would warn you that the certificate presented by this website does not belong to it site and would refuse to establish a connection.
In case of personal digital signature, certificate confirms that the owner is really the person described in the certificate fields. Otherwise everyone could claim to be Elon Musk or Jeff Bezos.
Validation depends on certificate type. For an email (an S/MIME certificate) and for basic domain validated certificate (a DV certificate) verification is usually simple and that's why relatively cheap. Validation of organization involves checking corporate registries and is more expensive.