They can also be approved to train for the IR but acclimatisation flying and the skill test must be done in one or more EASA member states.
The US schools can train up to the CPL, but the IR needs specific time in the EU.
The history is a long one dating to the days of the the Arnold Scheme under which seven British Flying Training Schools were set up in the US at Terrel TX, Lancaster CA, Miami OK, Mesa AZ, Clewiston FL, Ponca City OK, and Sweetwater TX. Other schools were created elsewhere in the empire and commonwealth—in Australia, Canada, New Zealand, South Africa, and Southern Rhodesia—under the Empire Air Training Scheme, Joint Air Training Scheme (South Africa), and British Commonwealth Air Training Plan. There’s some history about the BFTS here, http://www.5bfts.org.uk/articles/history/fullhistory.aspx
While those plans died out with the end of the war they must have paved the way for British civilian flying training in the late 1980s or early 1990s. To the best of my knowledge the first person to organise UK CAA approval for a US-based training school was Stephen Fisher at Flight Safety in Vero Beach, FL. He was followed by others like Ormond Beach Aviation set up by Adrian Thompson around 1992 if memory serves, and another at Kissimmee Municipal, now Kissimmee Gateway, managed by the Cranfield-based Cabair International, and Naples Air Center originally run by Nikki and Richard Gentil. One school running under IAA approval was a satellite of Waterford-based Pilot Training College. When it went into administration the training in Florida was reorganised by Florida Institute of Technology to be done directly under EASA oversight. FIT was the first organisation to achieve this feat at a cost of six to seven figures. I vaguely recall Sabena having a school in the US and it must have been approved under the Belgian CAA/DGAC.
Today training organisations providing EASA training outside the member states may still come under national approval if operating as satellite bases. Naples Air Center operates on this basis as a satellite of Andrewsfield Aviation in Essex. Otherwise enforcement and oversight must be done by EASA directly. The basis for this is article 78(1) of the Basic Regulation.
They have specific approvals from some EU countries. I think historically it was usually the UK which ran these. @tumbleweed might know more.
But there are EASA flight schools in the US ! How can they operate then ?
EASA license training could not be done anywhere in the world,
The training has to take place to a certain extent in an EU country, and the IRT itself administered in an EU country.
Well, yes, no doubt you are right, and the CAA does do a lot of rules which are outside its authority, but the practical reality is that if you are a flying school then the CAA “owns you” in the same way as every gun shop is “owned” by the local police chief 
There are procedures to force such things through but you just p1ss off the CAA and they will make your life hard next time they inspect you.
Actually I don’t see why EASA license training could not be done anywhere in the world, provided the instructor is correctly qualified and it is done through a correctly qualified school. So an FTO in say Berlin should be able to do a training flight to Kathmandu and back. It would be expensive for the student but very enjoyable and a hugely useful piece of flight training Anyone trying to ban this is just being silly. Anyway, what “through a school” really means? It’s nonsense. It’s a purely restrictive practice (which we have to live with in Europe).
Anyway it sounds like this UK CAA restriction has been removed fairly recently.
it does cost a lot to go to France just to do some training, so at the end only few business will go there no?
Depends where you are based. Those in the south east often fly to Lydd but their ILS is often booked solid. I know because I often fly it myself for practice (£20 if you don’t land). And alternatives like Calais and Le Touquet are very close… You would not do this if you are in Birmingham
It has always been the case with IR training, or IR tests, that you would start the day (or more likely the day before) with phone calls to nearby airports with IAPs, to try to book slots… It can be a difficult run-around.
Is such a restriction even legally enforceable? Training for an EASA license should be possible in the entire EASA airspace, no?
G-training goes deep in F-land, I heard things on the RT as far as Beauvais
Though who cares? at some point you will send to UK CAA a form with landings in France, if the specific licensing guy is OK you get the paper…
You may argue about “local schools business” but it does cost a lot to go to France just to do some training, so at the end only few business will go there no?
AFAIK a revalidation (which is possible with a freelance CRE/IRR etc) has never had such a restriction. Only the standard FTO-based initial IR training was thus limited, because the FTO had to get its ops manual CAA approved.
