Uniform margins are true of every copula, not just FGM copulas (not sure why that book leaves Gumbel out of the usual copula name).
In this case, uniform means "Has a uniform distribution" - that is, the density is constant over some interval, and in the case of copulas, where that interval is $[0,1]$, the corresponding distribution function is of the form $F_X(x)=x$ over that interval (and is 0 to the left of it and 1 to the right of it). Note that the cdf, $F_X(x)=x$ corresponds to a constant density, $f$, which is why the distribution is called 'uniform'.
Note that copulas have uniform $[0,1]$ marginals by definition. The particular copula you refer to has been chosen to fit with the definition. That it does so is easily seen; substitute $y=1$ into $H$ to see it for $X$, and $x=1$ into $H$ to see it for $Y$. Alternatively you can compute the marginals from the joint density by integration.
A search here for copulas will get you a lot more basic information relating to copulas.