After playing with this for a few hours I'd estimate the movement speed buff is between 1-5%, barely noticeable sadly.
It's possible that it only increases a chocobo's speed if you aren't already capped [First Class+Gallop], and the only speed I was noticing was lag. More testing is needed.
I've done some testing of my own for this, and I raced my fully capped (Full strength + gallop) with the silks, against my friend's fully capped with no silks. 5% speed boost seemed about right, as I was not that far ahead, but did run slightly faster.