How does the Franklin model explain a negative charge on an object?
Franklin believed in the "single fluid" notion of charge - one type of charge only. So, negative charge was deficit of a fluid.
In the early days of electric charge theory, the single fluid model was broadly accepted by many scientists. In this theory, there is only one type of charge, seen as a fluid that could be made to flow onto or off of objects.
If you made the fluid flow onto an object, that object now had a positive (amount of) charge, according to Franklin. If you caused the fluid to flow off the object, it had a negative (amount of) charge.
Franklin coined the terms positive and negative as though the words referred to "greater than zero" and "less than zero" charge. Nowadays, we do not see it this way, but "positive" and "negative" are still used, as they make the math manageable (don't they?).