There are two possible interpretations of 'I'll get it done'.
1. As Emma says, 'I'll get it done' can mean that you will arrange for another person to do it. This is often the case if you say this in a business context. You will make sure that the job is done, probably by instructing another person or people to do it. Someone else may do the work, but it will be your responsibility to make sure that this happens.
2. You can also 'get something done' by doing it yourself. If you use this construction, the suggestion is that it's a difficult or boring job that needs to be done in a certain time. For example:
Do you want to come to the beach this afternoon?
I'd love to, but I've got a lot of homework. I'll try to get it done this morning. If I manage to, I'll come and join you at the beach, ok?