Say your in the USA and want to watch BBC iPlayer and iPlayer will only allow UK based connections. If you try and connect straight from the USA you get bounced back.So if it's on your fire stick, how come the router in your house isn't detected as being in the UK? Don't you need a VPN on your router?
What the VPN software does is send your traffic to one of their servers based in the UK and from that server it connects with the iPlayer site. Hence the iPlayer site sees a connection coming from the UK rather than your physical geographic location. The site in question never talks directly to your router.
Device > Router > VPN Server > Site