Why the "meat" in "sweetmeat"?

The key is that meat used to just mean "food". The first entry under meat at Merriam-Webster has exactly that definition, though of course the primary sense of meat today is definition #2, which specifically refers to animal flesh. However, the older, broader sense of "meat" still sticks around here and there, such as the practice of referring to the edible part of a nut as the "meat", and in compounds such as "sweetmeat".

The OED reports the following definitions for meat (I report the first two):

1. Food; nourishment for people or animals; especially solid food, as opposite to drink. Now archaic and dialectal. OE.
   

   S. Johnson: The horses could not travel all day without rest or meat.
   Shelley: He had… meat and drink enough.
   Proverb: One man's meat is another man's poison.

2. A kind of food, an article of food. Obsolete except in sweetmeat. OE.