I agree with Doug. The second part of the sentence adds emphasis to the first. However, examining the grammar of the sentence might help.
“That” in the sentence in question is a relative pronoun, like “who.” “whom,” and “which.” Relative pronouns “relate” one part of a complex sentence to another.
For example, in the sentence, “He lives in the house that sits on the hill,” the subordinate clause, “that sits on the hill” relates back to the first part of the sentence and helps define which house.
When a sentence has two “that” clauses, it’s harder to interpret. For example:
“He lives in the house that sits on the hill that my family once owned.” Did the family once own the hill or the house?
Or take the old joke from the movie, Mary Poppins: “I met a man with a wooden leg named Smith.” Was the man named Smith or the leg?
When you start adding dashes or colons, the meaning can become even more subject to interpretation.
The sentence in question, “We admitted we were powerless over alcohol - that our lives had become unmanageable” could have been written:
“We admitted [that] we were powerless over alcohol [and also] that our lives had become unmanageable” or
“We admitted [that] we were powerless over alcohol [and therefore] that our lives had become unmanageable.”
Usually, however, the addition of a dash or a colon suggest that the last clause (i.e., “that our lives had become unmanageable) is a restatement of the first and is used for emphasis.
Break your sentence into its simplest parts and you get:
We admitted [something]
We were powerless over alcohol.
Our lives had become unmanageable.
When you put it back together and it becomes a complex sentence, the meaning also is more complex and therefore subject to interpretation. It could mean that an unmanageable life was the result of powerlessness, or that the unmanageable life was an example of powerlessness.
I don’t think that the meaning of “that” is the problem. Rather, I think it’s the dash between the two clauses that creates the potential for multiple meanings. I still think it’s a restatement for emphasis, however. It’s one problem (alcohol) with several results.