Here are some examples, and then I'll try to deduce a rule from them.
Who'll look after you, with me stuck in the hospital like this?
is equivalent to:
Who'll look after you, as I am stuck in the hospital like this?
You could use the word "since" (in the sense of "because") instead of "as." You could also in this particular instance reword it with the word "while" - i.e. "... while I am stuck in the hospital like this...," but this doesn't always work. Consider:
He was a great football coach. With him having been fired for
inappropriate comments, who will lead the team?
This is equivalent to:
He was a great football coach. As he has been fired for
inappropriate comments, who will lead the team?
"Since he has been fired" would also work, but "while he has been fired" wouldn't make any sense here.
So, to attempt to describe the rule, I would say that "with + object pronoun + description of the state of the object" is equivalent to "as/since + subject pronoun + verb construction of the state described." But this is just my impression of how it works and my own attempt to formulate a rule.