And so he did, however, the quote is out of context. Göring said this in a radio broadcast in 1933 when he was prime minister of Prussia. The quote refers to a newly enacted law banning vivisection of animals, it had nothing to do with sending people who ate meat to concentration camps. The quote, in full, was:

"An absolute and permanent ban on vivisection is not only a necessary law to protect animals and to show sympathy with their pain, but it is also a law for humanity itself.... I have therefore announced the immediate prohibition of vivisection and have made the practice a punishable offense in Prussia. Until such time as punishment is pronounced the culprit shall be lodged in a concentration camp."

This was an unsurprising move from the NSDAP, they had tried to introduce similar legislation two years prior in the Reichstag but failed to secure enough support for it (keep in mind they were still a minority in 1931). Animal welfare, particularly opposition to kosher slaughter and animal testing, had long been a part of the NSDAP platform. These were important political questions in early 20th century germany and there was a substantial and active animal welfare movement in Germany.

The total ban on vivisection was modified later the same year to allow for limited animal testing for certain institutions with a permit, provided that these were conducted with anesthesia when possible and followed ethical guidelines. However, there is some debate on how closely these laws were observed. In 1936 the German veterinary society lodged a formal protest against what they precieved to be widespread illegal animal testing.

Including this into their political platform was an uncontroversial move, there were quite a few animal welfare advocates and conservationalists among the top Nazi brass. Hitler was a strong advocate of animal rights, as was Himmler and Göring. I've always found this something of a paradox since these were the men who would become the main architects behind the holocaust yet Görings sent a fisherman to a concentration camp for cutting up a frog as bait.

Although somewhat rewritten and modernized (and with the punishment obviously adjusted) the animal welfare laws enacted in the 1930's are still largely the basis for germany's present day animal welfare laws. The NSDAP also introduced serious restrictions on hunting and created educational programs for both schools and hunting groups on the ethical treatment of animals and ethical hunting.

With conservationalism the story is somewhat different. While the NSDAP created several national parks and introduced restrictions on exploitation of nature there were a bewildering number of exceptions made in the name of Autarky. I highly recommend you take a look at /u/kieslowskifan's post here about environmental legislation in Nazi Germany and perhaps at the linked post by /u/commiespaceinvader in the same post.