Meaning
Wind of any strength — for in Patois 'breeze' covers everything from the gentlest draught to a strong gale, so 'di breeze a blow hard' can mean a real, buffeting wind, not just a soft air. As a verb, to 'breeze out' or 'breeze off' is to cool off, relax, or step out to catch some air, letting the wind take the heat off you. So the word names the whole range of moving air, from the pleasant coolness that relieves the heat to the fierce wind of a storm, and the act of going out to be refreshed by it.
Examples
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Di breeze a blow hard, hold on to yuh hat.
The wind is blowing hard, hold on to your hat.
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Open di window mek di breeze come in.
Open the window and let the air in.
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Mi a breeze out likkle before mi sleep.
I'm going to step out for some air before I sleep.
Origin
From English 'breeze', but in Patois it covers wind of any force, not just a light one.
How it’s used
Note it means wind generally, even a strong one. 'Breeze out/off' = cool off, relax, step out.