The mental or psychological recasting of a message so that its meaning is in line with a person's beliefs and attitudes, is

Mentally recasting a message so its meaning fits one's own beliefs and attitudes is selective perception.

Here the receiver reads into the message what suits established convictions, filtering it through personal bias.

Assertion means stating a view firmly and confidently, an expressive style rather than a reshaping of meaning.

Lobbying means seeking to influence decision-makers toward a cause, an act of persuasion, not internal interpretation.

Reflection means thoughtful contemplation, which does not by itself bend a message to fit prior attitudes.

It follows selective exposure, where people first choose messages aligned with their views.