"I am sick" is straightforward. It's like saying, "I'm currently experiencing illness."
"I happen to be sick" adds a little something extra. It's like saying, "I'm sick right now, but this isn't my usual state." It makes it sound a bit more like the illness is a temporary thing.
So, both mean you're not feeling well, but "I happen to be sick" adds a slight emphasis on the temporary nature of your illness.
Does that make sense?