After some searching, it turns out that this monster is very much a real chess problem. The official name for it is "The Obelisk."
The earliest known reference I have found is from February of 1882, mere weeks after its publication. Bizarrely, it comes from an Australian magazine. The site I found it on is trove.nla.gov.au.
It appears in the February 25th, 1882 edition of the South Australian Weekly Chronicle, published in Adelaide, on page 25. It is part of an overall review of Brentano's January 1882 issue.
Here is a clipping of the relevant section to read for yourself.
I also found mention of it in the "American Chess Review, Volume 1, Issues 1-6"readable on Google books. It is dated to 1886, four years later.
The information is given in the middle of the page. The position itself is given near the end of the page in descriptive notation. More about what that is on Wikipedia. For your information, the "S" is the German notation for the knight; it was semi-standard back then.
Here are later sources.
Chess Review 1936, issue 37 page 17; viewable online in the US Chess Federation's online archive
Curious Chess Facts, by Irving Chernev, 1937
Wonders and Curiosities of Chess, page 205, by Irving Chernev, 1974
The Complete Chess Addict, by Mike Fox & Richard James, 1987
The Even More Complete Chess Addict, by Mike Fox & Richard James, 1994
Unsurprisingly, there is no solution given anywhere at all. The reason for this is that the magazine ceased publication soon after the mate in 1220 was printed. Thus, another edition with the solution was never made.
As for how to interpret the stipulation, I believe it is very simple. White forces the Black knight to visit each and every square three times. Then White gives checkmates. Most of the moves come from White's pieces dancing around to do said forcing. If anyone is crazy enough to try and solve it, they will surely find a much shorter solution than whatever Babson had in mind.
However, when it comes to solving the problem, I have found a sequence that allows for all of White's pawns to be promoted. This is surely more than enough force for the whole solution for whosoever seeks to truly solve it.
[FEN "3nk3/3NN3/3PP3/3BB3/3PP3/3PP3/3PP3/2RQKR2 w - - 0 1"]
1. Nf6+ Kf8 2. Nh7+ Ke8 3. Qa4+ Nc6 4. Qb5 Kd8 5. Qb6+ Ke8 6. d7+ Kxe7 7. Bf6+ Kd6 8. Qa6 Kc7 9. e5 Kb8 10. Qb6+ Ka8 11. Qb5 Ka7 12. Ra1+ Na5 13. Bc4 Ka8 14. e7 Ka7 15. d5 Ka8 16. e8=Q+ Ka7 17. Qh5 Ka8 18. d6 Ka7 19. Be6 Ka8 20. d8=Q+ Ka7 21. Bc8 Ka8 22. e6 Ka7 23. d7 Ka8 24. e7 Ka7 25. d4 Ka8 26. d5 Ka7 27. d6 Ka8 28. e4 Ka7 29. e5 Ka8 30. e6 Ka7 31. d4 Ka8 32. e4 Ka7 33. d5 Ka8 34. e5 Ka7 35. Qh8 Ka8 36. e8=Q Ka7 37. Qeg8 Ka8 38. d8=Q Ka7 39. Qdf8 Ka8 40. e7 Ka7 41. e8=Q Ka8 42. Qeg6 Ka7 43. e6 Ka8 44. e7 Ka7 45. e8=Q Ka8 46. Qee6 Ka7 47. d7 Ka8 48. d8=Q Ka7 49. d6 Ka8 50. d7 Ka7 51. Qde8 Ka8 52. d8=Q Ka7