A saddled horse has a saddle on its back. A saddled person, on the other hand, is burdened or weighed down by some difficulty.
If you're saddled with debt after going to college, it means that the money you owe on your student loans is a huge hardship and responsibility for you. This figurative meaning of the adjective comes from the horse meaning — which is easy to see if you imagine the heavy saddle (and rider) being a burden for the poor horse that has to carry it around.