OOO, OOO, OOO, I got one of those routers. Gives whole new meaning to zoom zoom.
Ok the 1300mbps on the 5Ghz band is just ad hype. You'll never go that fast. Under ideal laboratory conditions, maybe.
BUT in order to get even close you'll have to have the matching wifi card or dongle. Look down at the bottom of this link.
Wireless AC - Wireless - Products - Welcome to TP-LINK
You'll need one of the dongle or internal card.
But to directly answer your question, there is a relation somewhat. If you can get the data stream out of the router into your computer and get it processed while maintaining that reference of speed, then it will be fast enough to be seen as magic. But if your connection between your computer and the router is slow, then it doesn't matter how fast your ISP is.
I'm hard wired in to mine. Gigabit on the computer and gigabit on the router. 40 down and 20 up. It's magic...
Going too fast will leave your computer waiting for the data to catch up.