Four days out of Puerta Ballena -- four humid, still days, with air so close you could cut it with your cutlass...
It has been hot and humid for four days, without any wind. The air feels thick like it has substance you could touch. There is a Tension as well, a feeling of expectation of a change. A somewhat common phrase in English, is "you could cut the tension with a knife". This expectation and the humidity are both sometimes refereed to by implying the air is "thick". Its entirely idiomatic.
...if Tinima weren't freshening the winds
I believe this would be said "as if Tinima weren't freshening the winds". I think Tinima is a river in Cuba, but they are anthropomorphizing the wind (pretending the wind is a person) and calling it Tinima. It just means that there is no wind.
-- there's a break in the heat. The cool natural breeze is at first a relief.
Then the wind moves. It feels nice, for a minute.
"Barometer keeps dropping," Caesar says.
A Barometer is a tool to measure air pressure. Weathermen use them. When air pressure drops quickly, there is likely to be a storm.
Tinima closes her eyes and breathes deeply. You can feel the tingle of her power stretching out beyond you. "There's a storm being born to our west."
The wind increases very much, and feels large and powerful. The people know that a storm is coming.