The expression register is not really a register itself, but it provides a way to evaluate expressions an insert the result into the buffer as if it had been retrieved from a buffer.
This is easiest to understand with a simple example. Starting with an empty buffer we enter insert mode:
Enter INSERT mode
iThe value of 2*pi is
The·value·of·2*pi·is·
INSERT
Top
1:22
then access the expression register:
Access the expression register
<C-r>=
The·value·of·2*pi·is·
COMMAND
Top
1:22
now we can use the expression register to calculate the value of 2*pi:
Calculate 2*Π
2*3.14<CR>
The·value·of·2*pi·is·6.28
INSERT
Top
1:26
The result of the expression (6.28) has been inserted at the cursor location.