The expansion rate of space is not itself the reason that the radius $R_mathrm{Uni}$ of the observable Universe is larger than 14 billion lightyears (Gly). Just the fact that space expands is the reason. If space did not expand, then $R_mathrm{Uni}$ would be the expected 14 Gly, as this is the distance that light can travel in the 14 billion years (Gyr) since the Big Bang. But since space has expanded in the meantime, then the distance to some particle that emitted a photon that we observe today, has been continuously increasing during the 14 Gyr, and hence $R_mathrm{Uni}>14,mathrm{Gly}$.
An analogy to this is a worm on a rubber band that is stretched while it crawls along. The distance between its starting point and it's end point is not just a matter of how fast it crawls, but also of how much you stretch the rubber band.
The term "space expands faster than light" is a bit deceptive. Space expands *homogolously*$^1$, meaning that a given point in space recedes from you at a speed depending on its distance from you. If a galaxy is 1 Mpc (= 3.26 Mly) from you, it recedes at $70,mathrm{km},mathrm{s}^{-1}$. If it's 2 Mpc from you, it recedes at $140,mathrm{km},mathrm{s}^{-1}$. And so on. At some distance the recession velocity becomes larger than $c$, and in fact galaxies at a distance of $R_mathrm{Uni}$ recede at more than $3c$. This doesn't violate special relativity which says that nothing can travel through space faster than $c$, because the galaxies do not travel through space. They lie approximately still in space, but space itself simply expands, i.e. the distance between everything increases.
Like three worms on a rubber band, with 1 cm between them. If you stretch the band to double length, the distance from worm #1 to #2 is 2 cm, while from #1 to #3 it's 4 cm.
$^1$Homogolous is not the same as homogeneous. Whereas the latter means "the same everywhere", referring to some physical quantity like density, homogolous means "proportional to distance".
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