I would guess the source is the first part of Appendix E, which states that the letter "i" represents the sound in "machine". The section's mostly about the Elvish languages, but there's a note that "in names drawn from other languages than Eldarin the same values for the letters are intended".
I should also note that S "is always voiceless, as in English 'so', 'geese'; the z-sound did not occur in contemporary Quenya or Sindarin." And for that matter, "e" should be as in "were", not a schwa. In a non-rhotic accent, I'd spell this pronunciation as "Eesserngard". (Edit: and the R from the original is thrilled! "Eesse(r)ngarrrd". The A at least is right, pronounced as in "father".)
But an actual recording of Tolkien definitely beats that!
EDIT2: Crikey, when you follow the Appendix E rules you get a very VERY specific accent. O is the sound in "for", so now pronounce "Rohan" and "Frodo" (remember to trill the R if you can!). I'm not even sure the text excludes "Samwise" from having three vowels: "father", "machine", and "were" respectively. Sahmweesseh.
I somehow don't think that's what Tolkien intended.
hS