1a I mean is often used as another way of saying for example. It precedes a list of things the speaker is serious about, things they feel the listener may find unbelievable, but that prove a point they're trying to make. Here, intonation is very important — both words are unstressed and pronounced with a very low, flat intonation. The nuance at work here is <...and I'm not exaggerating...>
Carl: So how was Julian's wedding?
Boris: The venue was nothing to write home about, but the food... what a spread! I mean beef Wellington, shrimp, lobster, caviar, pâté, the works!
1b I mean is often used to cite one clear example that proves a point you're trying to make. For instance, in the snippet below, Philip says Danny's cheap, and then to show just how cheap he is, he cites one instance of his cheapness that definitively prove his point beyond any doubt. In this usage, it has a nuance of <Here's an example you can't argue with:...>
Bob: So I asked my siblings to chip in so we could get my parents a large-screen tv for their anniversary. Natasha gave me $300. Tammy gave me $250, and Danny sent me all of $20!
Philip: And you're surprised?
Bob: What's that supposed to mean?
Philip: It means that your brother's a cheapskate. He's the richest and tightest person I know. I mean he gave his own wife a keychain for their anniversary, for crying out loud!
2 In another more abstract usage, unstressed I mean is used as a kind of filler, where it means something like <the thing is...> with a hint of <don't get me wrong...>. Here, the speaker's being cautious — and even a bit apologetic — about what they're about to say next. It's usually something that's not easy to say or pleasant to hear, but needs to be said in the speaker's opinion. In this context, it's often lengthened and pronounced with a hanging intonation:
Nicky: Dougie stayed over last night again?
Alex: Yeah, his parents are fighting all the time now, and he doesn't want to be there.
Nicky: I mean... I'm glad he's comfortable here, but he can't just like move in, Alex.
Alex: Why not?
Nicky: That's just not the way things are done. I like Dougie — I just don't want his parents to think we've taken him in.
3 Using the same hanging intonation, I mean... is often used to express doubt when presented with something meant to serve as proof. In this usage, it has a nuance of «Let's be reasonable/realistic!» followed by an explanation of why you don't feel the proof is conclusive. The speakers tone should be skeptical and drawn out to express a hesitation to agree with the proof presented.
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Conrad: Here, I found a picture of Jessica kissing some guy at a football game. Still don't believe me?
Alex: [taking the laptop and looking closely at the picture] I mean... that can be anybody — you can't see half her face and they're sitting in a shadow.
Barbara: Wendy called me this morning.
Nina: Let me guess... Bart beat her up again.
Barbara: Yep. When's she going to finally leave that idiot? Every time she takes him back, he does something worse than the last time.
Nina: I mean!