I was still puzzled from the answers, so I tried to calculate with all the steps. I take this definition "$C_k$ is the number of reads containing a k-mer." and corresponding definition for coverage ($C$): "$C$ is the number of reads covering a base".
Coverage is $C = \frac{T \cdot R}{L}$, where $T$ is total number of reads, $R$ is read length and $L$ is length of genome. Given the $C_k$ definition, $C_k = \frac{T (R - K + 1)}{L-K+1}$, where $R - K + 1$ is just number of kmers in a read, and $L-K+1$ is number of kmers in a genome. Then,
$$C_k = \frac{T (R - K + 1)}{L-K+1} = \frac{T (R - K + 1)}{L-K+1} \cdot \frac{R}{R} = \frac{R - K + 1}{R} \cdot \frac{T \cdot R}{L - K + 1}$$
since $L >> K$, we can approximate $L - K + 1 \approx L$, then we reduce the expression to
$$\frac{R - K + 1}{R} \cdot \frac{T \cdot R}{L} = \frac{R - K + 1}{R} \cdot C$$
which is the formula for $C_k$.