This is an example of what's called personification, which involves the attribution of human qualities and characteristics to inanimate objects. In this case, thirst—which is, of course, a recognizable characteristic of human beings who haven't had enough to drink—is attributed to an inanimate object, namely a front porch swing.
In this particular extract, Woodson's describing her grandparents' house. The front porch swing is described as "thirsty" to emphasize the fact that it's in desperate need of some oil. This would appear to suggest that it's been a long time since the swing was used; by children, for example. As Woodson's grandparents clearly aren't in the habit of using the swing themselves, it just sits there on the front porch, waiting for some unspecified moment in the future when children will sit on it once more.