It's the way you look at it:
The word "traffic" is uncountable. We say, "There isn't MUCH traffic at 5 in the morning." We can also say "There aren't MANY cars in the street at 5 in the morning."
Some words can be both:
"You won't need MUCH material to finish the coat."
"A patchwork quilt is made of MANY materials."
A lot of uncountables can be singularised by using "a piece", not only things which can be physically broken up like "a piece of cake" or "a piece of rock" but abstract nouns like "I need one more piece of information." which emphasises its singularity from "I need some information".
In the case of "bread" the word is usually singularised by the word "loaf". "I want some bread, two loaves please." (although a slice of bread, a piece, etc. are OK of course, in the right context). It can, however, be pluralised in the right context - very specifically about different types of bread. Google it and you will find phrases like "Breads are a group of staple foods."
Meat can be singularised in two ways:
Mutton and beef are two KINDS of meat.
I like smaller PIECES of meat in a ragout.
It can, however, in a clear context, also be used in the plural, as you will see if you google the word. I have just picked, "Other Meats, including rabbit, ostrich, and frog's legs..." off a website.
However in old English the word "meats" meant rather "dishes" (from the French "mets" than meat in its modern sense and in some contexts a degree of ambiguity might subsist. Consequently, even if "meats" can be used, avoid it.It will probably sound funny. Use "kinds of meat", "pieces of meat", "joints of meat" as appropriate instead.