As indicated in some of the other answers/comments, similarity between languages can be compared in terms of many different factors. In terms of sounds, Esperanto might be said to be similar to the many languages that make a 5-way vowel distinction, such as Spanish and Japanese. However, Esperanto allows more complicated syllable structures (e.g. skribi has 3 consonants, s-k-r at the beginning of a consonant;syllable; this particular combination isn't allowed in Chinese, Japanese, Korean, Spanish...) relative to those two languages.
Similarly, Esperanto might be called similar to English, Chinese, French, and other languages in allowing (and in practice, preferring) a Subject-Verb-Object word order:
Mi amas esperanton.
Li skribis leteron.
Vi vidos librojn.
But Esperanto might be called similar to Japanese in overtly marking direct objects:
boku-wa hon-wo yomi-masu
I-TOPIC book-DIRECT OBJECT read-POLITE
mi libro-n leg-as
I book-DIRECT OBJECT read-PRESENT
Depending on what aspect of Esperanto we are looking at, Esperanto can be considered similar or dissimilar to many of the world's languages.
Really, it is difficult to say what language Esperanto is closest to grammatically because grammar is a complicated thing consisting of many layers of analysis (e.g. syntax, morphology, phonology, etc.). However, we can clearly look at some one feature of Esperanto and see how that feature compares to other languages of the world.
