Psychiatric hospitalization is recommended when people are in imminent or immediate danger of hurting themselves or others in some way. So, if someone with schizophrenia is having suicidal thoughts, having a lot of hallucinations, or hearing voices telling them to hurt themselves and they feel they can't manage it, then that's a good time to consider hospitalization. If they are agitated and aggressive, as well, and at risk of hurting someone else, in the cases of feeling very paranoid that someone's out to get them and they need to defend themselves, or hearing a voice telling them to hurt someone else, that might be another good time to seek hospitalization.
Then, there's poor self-judgment and poor self-care. Sometimes, someone with schizophrenia, for instance, is totally unable to manage their hygiene. They may hoard things to a dangerous level where there are vermin and they are in danger of developing terrible diseases or needing serious medical care if they continue in that situation. Then, that's also a reason for hospitalization.
For some people with schizophrenia, using a lot of drugs or alcohol to a degree that is dangerous or that is not remitting, this may be a reason why they might choose to be hospitalized in order to detoxify and seek rehabilitation.