(Apologies, I'm writing this comment in a way so that even passersby who have zero knowledge of Esperanto can make more sense of it)
As for the research, honestly there's quite a few very easily findable online, especially via Wikipedia reference links on various pages related to Esperanto. The "open library" has a few books with research summaries in them at least, one being here. The one problem is that when you're reading only in English, they tend to only be looking at how Esperanto helps with Romance languages (because that's all English-speaking countries tend to care about, not because Esperanto only helps with those). If you know anyone who speaks Esperanto, it's much easier to find the research on it through that, and there are also relatively a lot of studies on it done by France so try searching in French if possible. (There's even been notes, if not exact studies, in Swedish among other languages). While I've read a lot of reports, I haven't kept the links to any of them, but I found them by simply searching for general information about Esperanto. Any remaining research reports basically need to be actually purchased from somewhere (ex. a University).
Here's just a couple:
Modern research, date unknown, better sources for it must be able to be found: About how the Chinese counting system improves mathmatics grades, among other things. Esperanto uses the same exact counting system, the words are just a tiny bit longer to say: http://gladwell.com/outliers/rice-paddies-and-math-tests/
2012-2013: Various reports from Australia on the effects of using Esperanto in primary school as a first second-language for the children: http://www.esperantoresearch.org.uk/site/publications
Since I haven't saved any more links, here I will show you some examples I've personally gathered, for why Esperanto helps with learning other languages. It's not exact scientific research but it ends up being pretty obvious in showing how Esperanto helps, I think anyway.
I have to first summarize the main idea, by saying that in Esperanto, word-construction (compounds, affixes), grammar markers (anger vs angry vs. angrily; accusative case), certain words (reflexive pronouns), and so on are simply the basics of grammar of most languages, especially Indo-European ones. It's the same as going through a page of a book, and colour-coding which words are adjectives in one colour, which words are nouns in another colour, which are direct/indirect objects in another colour. That is partially what greatly speeds up the learning of other languages; it very, very clearly teaches you when to use each form of a word in a sentence, whether it's in Esperanto or any other language.
Especially compared to English, Esperanto has a small base vocabulary with few synonyms, which very closely mirrors languages like Japanese and Swedish; the remaining necessary words are created by compounds in all three of these languages. That's another main reason for why Esperanto speeds up learning, because the English-speaker's brain is already trained to use "only basic words", as must be done in most foreign languages.
Now for some explicit language comparisions so you see what I mean:
1. Vocabulary and word make-up (forgive me for not bolding the relevant parts, I'm too lazy):
Esperanto (EO): kat-ido, hund-ido, ĉeval-ido
Swedish (SV): katt-unge, hund-unge, häst-unge
English (EN): "baby cat, baby dog, baby horse", kitten, puppy, foal
(EO) manĝ-voli, tranĉ-ilo, vet-eto
(Greenlandic) neri-uma, savi-mmik, eqquiniu-nnguaq
(EN) "eat-want, cut-tool, wager-ette", to want to eat, a knife, a small bet. Note that in Japanese instead of cut-tool, "small sword" is said.
2. Accusative case:
(EO) ĉi tie, ĉi tien, hejme, hejmen, ekstere, eksteren, li, lin
(SV) här, hit, hem, hemåt, ute, ut, han, honom
(EN) here, hither, home, homewards, outside, out, he, him
———— (EO) La kato kuras sub la sofo
(Faroese, FO) Kettan rennur undir sofuni
(EN) The cat runs around under the sofa (without leaving the underside of it)
———— (EO) La kato kuras sub la sofon
(FO) Kettan rennur undir sofuna
(EN) The cat runs underneath the sofa (it runs from outside of the sofa, to underneath it)
3. Reflexive pronoun difference: (I don't know French etc. but I'm fairly sure they have the exact same difference)
(EO) La knaboj ludas kun sia kato
(FO) Dreingirnir spæla við sína kettu
(SV) Pojkarna spelar med sin katt
(EN) The boys play with their cat (their own cat)
———— (EO) La knaboj ludas kun ilia kato
(FO) Dreingirnir spæla við teirra kettu
(SV) Pojkarna spelar med deras katt
(EN) The boys play with their cat (a cat owned by someone else)
———— (EO) Faru ĝin mem. Li ŝatas sin. Ĉesu vundi sin mem
(SV) Göra det själv. Han gillar sig. Sluta skada sig själv
(EN) Do it yourself. He likes himself. Stop injuring yourself
As you can see, even in non-Indo-European languages such as Greenlandic and Japanese, there can be some direct similarities to Esperanto. This definitely isn't all the similarities but it's just a a taste. It's really hard to describe most of the similarities in just a few words or examples since English is just too different, plus I don't want to go into the grammer etc. of Japanese and so on, so I don't want to go on any further. That includes giving better examples of compound words among other things. Sorry about that, I'm just too lazy. Of course, every similarity is what helps you with a foreign language.