Literal translation
he believes he has arrived
Meaning
he thinks he's made it
Explanation
This expression describes someone who takes himself seriously, convinced he has reached a success or a standing he has not actually attained. The verb used carries the idea of growing, of inflating one's own importance in one's own eyes, while being arrived means having succeeded socially, having reached a certain rank. The expression plays on the gap between how the person sees himself and the more modest reality of his situation. The tone is ironic and mildly mocking, reserved for familiar conversation among people who comment, often with a smile, on someone's vanity or self-satisfaction. It is close to French turns such as se croire arrivé or avoir la grosse tête. The register stays casual, without coarseness, fit for any situation where one wants to point out, without excessive unkindness, that a person overestimates his own path or social importance.