If you do something “gingerly,” you do it cautiously or carefully. This makes sense in this context, and Andrew is trying not to hurt himself, and adjusting the manner in which he dismounts the fence accordingly. Do not confuse it with, "gingery," which means related to the spice ginger. To clarify the other terms, “reddening” means turning red, such as when blushing; a “fastening” is something you use to affix things together; “hastily” means hurriedly or quickly.