I can only speak for the origins of football/soccer and the sports that have derived from it, but before I delve into the history, I guess I would say for the sake of the joke in the meme, the claim is true. It says in the meme, "Invented sport" and they are referring to soccer/association football in this case as that badge is of the english football association, and the rules were codified in england and now they're national team is not that great considering the lofty expectations. So to answer your main question, it is a valid claim despite being a generalization.

Delving into the history part, England did not "invent sports." Sport has been popular for a long time (I am sure you have heard of the ancient olympics). In Europe, however, sports played with a ball, despite existing, had no great status, things like chariot racing, archery, athletics, and wrestling were the dominant sports of classical antiquity. In other places, there are records of different games played with a ball, such as cuju (translated as kick-ball) dating back to the Han Dynasty (206 BCE-221 BCE) in China, or some ball games discovered by the Spanish upon arrival in the Americas. So no they did not event sport or even ball games played with your feet.

However, they did invent modern "association football" (the official name of soccer), even if a history of ball games throughout human civilization made the world predisposed to accepting it. Like many other things, this is largely a product of Britain being the first to industrialize. In the 18th century, a big sporting culture was developing in Britain, and this snowballed in the 19th century with increased industrialization and urbanization. There are a lot of details and debate over the specifics, but basically what happened was different versions of a similar game with an equal number of men trying to score points at either end of the field with a ball was emerging throughout the UK. As popularity increased especially amongst british public schools, people tried to codify the rules of the game, for at the time each region had their own version. There was a lot of debate over this, with the main point of contention coming from whether people should be able to use their hands. This eventually split the two different sports, with one becoming football/soccer and one becoming rugby. The codification led to the first FA Cup to be played in the 1871-2 season. It gained popularity and as the British empire expanded, it helped spread the sport across the globe. I don't know much about tennis, hockey, or cricket, but American Football derived from some sort of intercollegiate version of soccer before it had been fully differentiated from rugby. Also note that soccer in these early days was much more physical and there was not as much of a physicality gap between rugby/American football and soccer today.

My source for all of this is the fantastic book, "The Ball is Round: A Global History of Soccer" by David Goldblatt. I highly recommend it, although it is very long

edit: fixed some typos and clarified some grammar to make it clearer

At what point does an idea turn into a sport though? I'm most familiar with hockey and have done a lot of reading on the history of it. There had been many stick and ball games before it, but at least in the case of Ice Hockey the English were the first to create rules for the game. Most people have always assumed that British soldiers in Canada standardized the game. However, recently there was a book written by hockey historians claiming the game can be traced back to England and that Charles Darwin made mention of it long before anyone in Canada laid claim to creating it.

The book is "On The Origin Of Hockey"

http://www.amazon.ca/On-Origin-Hockey-Carl-Giden/dp/0993799809

I have not read it yet, but I know it's all the buzz within the hockey world right now.