What does it mean to say that the gravity of the Earth is 9.8 m/s2?

It means that any object is attracted by towards its center with a Force ##F=mtimes g##, where ##m## is the mass of the body and ##g## acceleration due to gravity, stated in the question.
As per Law of Universal Gravitation the force of attraction between two bodies is directly proportional to the product of masses of the two bodies. it is also inversely proportional to the square of the distance between the two. That is the force of gravity follows inverse square law.
Mathematically
##F_G prop M_1.M_2##
Also ##F_G prop 1/r^2##
Combining the two we obtain the proportionality expression
##F_G prop (M_1.M_2)/r^2##
Follows that
##F_G =G (M_1.M_2)/r^2##
Where ##G## is the proportionality constant.
It has the value ##6.67408 xx 10^-11 m^3 kg^-1 s^-2##
##r## is the mean radius of earth and taken as ##6.371 times 10^6 m##
Mass of earth is ##5.972xx 10^24 kg##
If one of the body is earth the equation becomes
##F_G =(G (M_e)/r^2).m##
See this has reduced to ##F=mg##
Were ##g=G (M_e)/r^2##
Inserting the values
##g=6.67408 xx 10^-11 (5.972xx 10^24)/(6.371 times 10^6)^2##
Simplifying we obtain
##gapprox9.8 m//s^2##
In other words if an object is dropped from a height ##h## above the earth’s surface, the object will fall towards earth with constant acceleration of ##g=9.8 m//s^2##