I'm a bit confused between the difference between scaling and centring in a negative binomial regression.

My model looks like this:
invertebrate abundance= Year +shoot count + epiphyte biomass(g) + epiphyte bio*shoot count

Years: 2015, 2016
Shoot count range:1-30
Epiphyte bio range: 0-2g

Centring involves subtracting means from each variable so the overall mean is 0. I have centred my continuous and categorical (Year) input variables, so that when I run a negative binomial regression, my coefficients can be interpreted as:

Each unit increase in x1 (shoots) increases abundance by exp(coefficient x1) when x2 (epiphyte biomass) is held at its mean (0), and x3 (Year) is held at an imaginary mean category (because I am interested in the average affect across different years, not the effect of year itself).

Because epiphyte biomass and shoots are on such different scales, it has been suggested to me to scale them (divide by their standard deviations).

Theoretically I know normalizing (scaling and centring) allows me to compare the effect size of the coefficients directly from the model- ei it will tell me which variable has a stronger effect on total abundance. But logistically I don't understand how this works, and how it changes the interpretation of my negative binomial model.