In your course or book, something that looks more or less like $\Delta N_{t\;\text{to}\;t+1}$ has been defined as the difference between $N_{t+1}$ and $N_t$, that is, as $N_{t+1}-N_t$. This is the change in population from time $t$ to time $t+1$. (A more common notation for the same thing is the more compact $\Delta N_t$.)
We have $N_{t+1}=\lambda N_t.$ It follows that $\Delta N_{t\;\text{to}\;t+1}=N_{t+1}-N_t=\lambda N_t -N_t=(\lambda-1)N_t.$
Thus $\Delta N_{t\;\text{to}\;t+1} =(\lambda-1)N_t=RN_t$, where $R=\lambda-1$.
Note: Sometimes people have trouble with the fact that $\lambda N_t -N_t=(\lambda-1)N_t$, even though they would have no trouble seeing that, for example, $\lambda N_t-5N_t=(\lambda-5)N_t$. We need to observe that $N_t=1\cdot N_t\;$:$\:$ If this year's population is $N$, and next year's population is $17N$, the change in population is $17N -N$, which is $16N$.