Is there a special app that people can download if they are outside of the country so they can vote on this years series

No, it's the normal X Factor app, they just use a UK Google/iTunes account and access it via VPN.

If it's anything like trying to get Pokemon Go before it was released in the UK then it's a huge pain in the bum and very few will have the inclination to jump through the hoops for a meagre five votes.

So I guess to answer the question in the title: They don't.

Tried myself to do so a week ago, but COULD NOT. Makes me laugh people here try to convince Saara's success was because of Finnish votes, simply .. not true.

Downloaded it in spain and voted from there

They don't. It's a silly rumour spread about to stop British people voting for her on the basis that "she's 'cheating'" already by having Finnish votes.

If you paid for an internet account and provider in the UK, from Europe, then yes I guess it might be doable, but really, who would go to that effort just to get five 'free' votes in a British talent show? You might as well pay someone in the UK to ring up and vote on your behalf.

they made it up about Saara getting votes from Finland, cheap tactics to make Matt the winner

I didn't think Finland had voting access, hope many people there voted for Saara