The default mode network (DMN) is a collection of brain structures identified in human functional neuroimaging.
Specifically, a type of neuroimaging called BOLD indirectly measures local metabolism in the brain; the assumption is that areas with a strong BOLD signal are active.
It's typical to use Pearson correlations of the BOLD signal over time between different brain regions to measure what is called "functional connectivity": regions are said to be functionally connected if they have high metabolism at the same time, and low metabolism at the same time. In other words, if their BOLD signals are positively correlated, they are said to be functionally connected.
The DMN is a particular collection of brain regions found to have high correlations among each other = high functional connectivity at rest, when people sit in the scanner doing nothing else. This is an observation that is very consistent and reliable in "normal" participants in fMRI studies.
Given this reliable signal, scientists have sought to connect the DMN to psychiatric disorders. If someone says "DMN hypoconnectivity in (condition A)", what they typically mean is "we measured Pearson correlations of BOLD signals in brain regions that people know are usually highly connected at rest; we found the strength of those connections was weaker in (condition A) than in participants without that condition". Hyperconnectivity would mean the opposite: higher correlations in (condition B) than in a control population.
Those are both examples of intraconnectivity in terms of the DMN: connections within=intra that network. It's possible to break the rest of the brain into other networks, or just one broad "not-DMN" network. If a paper talks about "interconnectivity" with the DMN, they are talking about the relationship between parts of the brain that are in the DMN and other parts of the brain.
Everything that comes after that is pure speculation, or needs to be supported by something else. In one of the papers you link, the authors write:
Constant overengagement of the default network could lead to an exaggerated focus on one's own thoughts and feelings as well as an ambiguous integration between one's own thoughts and feelings with events in the environment. Thus, neutral events would seem to be imbued with exaggerated self-relevance, and the boundary between the internal world of reflection and feeling and the external world of perception and action would be weakened. Indeed, many symptoms of schizophrenia involve an exaggerated sense of self-relevance in the world, such as paranoid ideation that individuals and groups are conspiring against the patient, and a blurring of internal reflection and external perception, such as hallucinations.
That's all just the authors' speculation (which is perfectly appropriate for a discussion section; that's what they're doing, discussing the results). They are interpreting the result they found (DMN hyperconnectivity) in terms of other things they already know about the psychiatric condition they are interested in.
