Why do we say that we model the rate instead of counts if offset is included?

I am fitting the following GAM model. My response variable is aggregated counts of disease incidence at the county-level. I used log(population) as an offset. If I use it this way, I learned that it models the rate of cumulative cases per unit of population while still treating it as a count GAM. However, I am not very confident in stating this in my manuscript. Could anyone help understand it better? cases ~ α + offset (log(population)) + ƒ(age ) + ƒ(school) + ƒ(associate degree) + ƒ(income) + ƒ(f_over) + ƒ(housing) + ƒ(overcr) + ƒ(smo) + ƒ(drink) + ƒ(dia) + ƒ(lung) + ƒ(obe) + ƒ(uninsured) + ƒ(flu vac) + ƒ(C) + ƒ(N) + ƒ(O) + ƒ(SO) + ƒ(cci) + ƒ(pre) + st + county(random effect) + ε In the partial effects plot, should I write the y-axis as incidence rate or just incidence? Similarly, in the main text, should I use the incidence rate, or just incidence would suffice, when explaining the effect of covariates on the response? As suggested by @Demetri Pananos, I developed the plot with the following code: dg $no), max(lc_data$ no), length.out = 50), pop = 100000) plot_predictions( model = gam_full, newdata = dg, by = 'no', type = 'response' ) Is this fine? I doubt the code above using the plot_predictions function from the marginal effects package. My doubt is whether I should write the code in this way (below) to get the population level of estimates excluding random effects (county is my random effect). Is this the right approach? dg $no), max(lc_data$ no), length.out = 50), po…