How do the terms in 千里之行，始于足下 modify its meaning?
Actually your translation is a pretty verbatim one. I will give the word for word translations for each phrase and let you feel what parts they play.

千=thousand

里=miles

之=of

行=journey

始=start, begin

于=at, from

足=feet

下=under

Something to comment on: in chinese, “A之B” is somewhat equivalent to "B of A", meaning B is part of A, or B is described by A.