How does curly brace expansion work in the shell?

They are called brace expansion.

It is one of several expansions done by bash, zsh and ksh, filename expansion *.txt being another one of them. Brace expansion is not covered by the POSIX standard and is thus not portable. You can read on this in bash manual.

On @Arrow's suggestion: in order to get cat test.pdf test.pdf test.pdf with brace expansion alone, you would have to use this "hack":

#cat test.pdf test.pdf
cat test.pdf{,}

#cat test.pdf test.pdf test.pdf
cat test.pdf{,,}

#cat test.pdf test.pdf test.pdf test.pdf
cat test.pdf{,,,}

Some common uses:

for index in {1..10}; do
   echo "$index"
done

touch test_file_{a..e}.txt

Or another "hack" to print a string 10 times:

printf -- "mystring\n%0.s" {1..10}

---

Be aware that brace expansion in bash is done before parameter expansion, therefore a common mistake is:

num=10
for index in {1..$num}; do
   echo "$index"
done

(the ksh93 shell copes with this though)

PesaThe's answer answer covers important aspects of the question. There are several things I want to add.

The asterisk in ls *.txt is handled by the shell and is therefore controlled by shell options which can be changed by shell built-ins. In this case, one can disable asterisk expansion by running set -f and enable it again by set +f.

Another thing is that anyone who wants to make the script portable should check the POSIX standard. For example {1..9..2} expands to 1 3 5 7 9 in bash 4 but does not expand in lower bash versions or in sh.