The phrase implies a task takes a long time and lots of effort to complete. She's been slogging it out for months to become proficient in English.
Pro Tip 2/3
Persistence
It's used when someone is persistently working hard without giving up. Even though the startup struggled initially, the team slogged it out and found success.
Pro Tip 3/3
Challenging effort
The phrase connotes that the effort required is strenuous and possibly draining. The mountaineers slogged it out against the storm to reach the summit.