Let's first get some more context from the excerpt and highlight a few words:
It’s always raining there, and there’s hardly a breath of air. I travel to Pachuca to see her once a month, usually on Sundays. But I never go as far as the cemetery, because I’m allergic to pollen and there are lots of flowers there. I get off the bus not far from the gates, at a lovely median strip with life-size dinosaur sculptures, and I stay right there among the gentle fiberglass beasts—getting soaked, saying Our Fathers—until my feet swell up and I feel tired. Then I go back across the street, carefully dodging the puddles—round as the craters in my childhood notebook—and wait for the bus to take me back to the station.
The context tells us that it often rains in that area; afterwards, the narrator also avoids the puddles (small pools of water) that the rain has caused.
The verb soak has several meanings that are relevant here:
- To be saturated with liquid by being immersed in it.
- To immerse in liquid to the point of saturation or thorough permeation.
- To penetrate or permeate by saturation.
- To allow (especially a liquid) to be absorbed; to take in, receive. (usually + up)
Sponges, floorcloths and towels are objects that can absorb or soak up water. The result is that they are soaked. The human body does not soak up water so easily, due to the skin. When a person says that they were getting soaked when walking through the rain, the rain got through their clothes and reached their skin. So getting soaked is an exaggeration or hyperbole. The expressions soaked to the skin and soaked to the bone also mean "extremely wet".