This is an example of some well written dialogue! When I was dissecting the meaning of "stripped bare" in this context, I found it useful to look at the individual meanings of each word and them look at the phrase as a whole.
The verb to strip means to remove all coverings ("stripped" is the past tense and past participle). Bare is an adjective that means exposed to view or basic and simple.
Looking at the phrase as a whole gives me a mental image of "truth" appearing complex and hidden from view because it is beneath layers of little lies. In the Hospital, the environment and the interactions of the people causes truth to be stripped of all nonessential aspects (little lies) until no layers remain and truth is exposed, and appears simple or "bare".
Stripped bare means that ALL superficial coverings or objects that are obstructing the basic nature of an object from view are removed. Here are a couple of less poetic and abstract examples of the phrase that come to mind: We have to strip the car bare, the flood has damaged the interior as well as the exterior. When his story was stripped bare the police were able to see that he was clearly guilty.
I hope that was a helpful for you!