Truncating ciphertexts on ring-LWE schemes
The expression $c' = \lfloor w^{-i} c \rfloor$ is a slight abuse of notation. What it technically means is to interpret each coefficient of $c$ as real number, divide by $w^i$, round the result to the nearest integer, and then interpret it as an element of $\mathbb{Z}_q$ again.

This is equivalent to expressing each $\mathbb{Z}_q$ coefficient of the ciphertext as a bitstring, and then discarding the $i$ least significant $w$-words of each bitstring. It also explains why $w^i c'$ is equal to c with the $i$ least significant $w$-words of $c$ being set to $0$, as the authors say in the next sentence of Section 5.4.

By the way, $w$ does not represent the number of bits in each word. Instead, $w$ describes the size of each word, and so the number of bits in each word is actually $log_2(w)$.

If you're interested in other examples of where the technique of discarding the lower-order bits is used, see works on the Learning with Rounding problem. (The paragraph in the middle of page 2 gives an equivalent explanation of this notation.)