Zdravím všechny,

nedává mi moc smysl věta z jednoho výukového materiálu. Dám sem předchozí kontext:

Well, I got dumped today. By my mom friend, Sandy. Apparently, she doesn't want to see me anymore.

That sucks.

It does, doesn't it? I couldn't even hang on to a mom friend. And it's not like she's all that interesting herself.

Right. I've met Sandy. She didn't make of an impression on me. Who does she think she is, telling you to stay out of her life?

But you know what, she's right. I am kind of boring to talk to. Every single one of those other moms is probably more fun to be around than I am.

Uh huh.

Are you not going to say anything? Do you not care?

Frankly, right now, I don't know that I do care. About that, I mean.

I can't believe what I'm hearing. How can you not care?

Jaký je prosím smysl zvýrazněného úseku?

The phrase is ok. The stress here is on „do“, so „I don't know that I do (care)“. The person was asked „Do you not care?“, and it's a slightly hesitant way for the speaker to agree that (s)he doesn't care. The word „care“ could be left out. The speaker could equally have said No, I don't think I (really) care at all, or No, I guess I don't (really) care at all. Maybe: co já vím, mně to asi nevadí (if that sounds idiomatic in Czech 😉 ).

Another example:

„Do you know the Name of God?“ asked the teacher.

„I don't know that I do.“ [= No, I don't think so ] (bible.com)

„I don't know that …“ can also be used to „soften“ the negation of the following verb, e.g.

• Honza has asked me out on a date, but I don't know that I want to go out with him because he drinks too much and he swears a lot. [= … but I don't really want to ..]
• I don't know that I agree with you = I don't really agree with you or I don't think I agree with you or I can't really agree with you.