Define the
variation of a functional ##F[\phi(x)]## as ## \displaystyle \delta F[\phi] = F[\phi + \delta \phi] - F[\phi] = \int \dfrac{\delta F[\phi]}{\delta \phi} \delta \phi \mathrm{d} x## the ##\dfrac{\delta F[\phi]}{\delta \phi}## is the
functional derivative of the functional ##F[\phi(x)]## w.r.t the field ##\phi(x)##.
Now construct a
localized variation of the field ##\phi(x)## at the point ##x=x'## with magnitude ##\epsilon##. This means that we can write ##\delta \phi(x) = \epsilon \delta(x-x')##.
Insert this in the definition of functional variation:
## \displaystyle \delta F[\phi] = F[\phi +\epsilon \delta(x-x')] - F[\phi] = \int \dfrac{\delta F[\phi]}{\delta \phi} \epsilon \delta(x-x')\mathrm{d} x = \epsilon \dfrac{\delta F[\phi]}{\delta \phi}##
This resembles the ordinary relation we have for a variation of a function ##f(x + \epsilon) - f(x) = \epsilon \dfrac{\mathrm d f}{\mathrm d x}## which is valid in the limit "small" ##\epsilon##.
Your book (well I also own it so I guess it's my book too!) then motivates the defintion of the functional derivative as
##\displaystyle \dfrac{\delta F[\phi]}{\delta \phi} = \lim_{\epsilon \, \to \, 0} \dfrac{F[\phi +\epsilon \delta(x-x')] - F[\phi] }{\epsilon}##
(I used ##\phi## insteaf of ##f## here) (note: you had some errors in the way you wrote down how the book defined the functional derivative...)
Let's pray a real mathematician chimes in at some point
TIP: if you really want a good QFT book get
"Field Quantization" by Greiner. It might be hard to get, but sometimes you get get it very cheap used. It covers only field quantization methods in an entire book about 350 pages. This is what most QFT books spend on 2 chapters worth of 20-30 pages.