about to go crazy; close to the edge
Describes a state of being on the verge of losing one's mental composure or control, often due to stress, frustration, or overwhelming emotion. The phrase is a set expression combining どうにか (somehow) with なりそう (seems like it will become).
I'm so busy with work that I feel like I'm about to lose it.
Listening to his story makes me feel like I'm going crazy.