1.
to feel proud; to feel one can hold one's head high among others
To feel free of embarrassment in front of other people, usually because of an achievement — often someone else's, such as a family member's. Literally 'one's shoulders are wide.' Far less common than its opposite, 肩身が狭い.
My daughter got into the school she wanted, so now I can hold my head high even in front of the relatives.
Structure: 肩身 (the shoulders; one's standing in others' eyes) + が + 広い (wide).
Common patterns
Usage
The pride is social — how one appears to relatives, neighbors, or colleagues — not private satisfaction, which would be 誇らしい. The negative counterpart 肩身が狭い (to feel small, ashamed) is much more frequent in everyday speech, and 肩身が広い is often used precisely as its reversal.