The original, classified information could be only handled inside a SCIF, on computers that are installed there and connected to whatever secure network they may be connected to, through the encryption hardware. One would presume that the State Dept. has a good number of SCIFs, as most of what they do is expected to be classified in some way.
If we presume that the secure computers don't have USB ports or other I/O that is suitable for exfiltration of information, then the criminal could smuggle a cell phone into the SCIF and capture the screen with a camera, or just retype relevant parts of the message into the phone. This would also remove classification markings, as they would not be copied. Naturally, phones and other such equipment is not allowed into a SCIF. But who is going to search high-flying officials of the department?
The easiest way to find out who sent the stolen classified information is by looking at the headers of those emails. They say something like "from:jane.doe.123567@abc.def.us.gov" - and you can take it from there. Given that there are many such emails, the senders are going to die in prison unless they make a deal with prosecution - and that means revealing the whole scheme. Managers are well known for insisting on their own IT solutions, but here the senders had to commit an obvious federal crime. What were they thinking?
Huma Abedin was on staff at the state department and had high level clearances. Couldn’t she have accessed her state department email and somehow copied the classified documents, then sent them to Hillary from a non-gov account?
Another plausible explanation is that Hillary had a flunky at state who sanitized the documents so that they could be sent outside the SCIF.
I think you could probably remove classification tags and markings by selectively copying content from a classified email and pasting it into an new, unclassified email.