I understand that this model of food means that the amount of food available decreases as the population increases, however I do not understand the two parts underlined in green. How do these agree with one another? Here I believe $F_0$ is $F$ at $t=0$, hence we must have $N(0)=0$
The wording is ambiguous but it makes sense if you take $F_0$ as the rate of food production, then $F(t)=F_0-eN(t)$ is the rate of growth in food stock at time $t$ corresponding to population $N(t)$. Then you will get a carrying capacity $N_{max}=F_0/e$, the population level at which all the food production is consumed.