(Inspired by this question:) Which natural language is closest to Esperanto gramatically (accusative as the only case, future tense of verbs, verbs not inflected for person etc. etc.)?
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 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.
Verbs don't conjugate, there is basically only one past tense, etc.
This is a rather broad question, and different parts of the grammar will lead to different answers here. With respect to the treatment of the definite article, it turns out that central Semitic languages (like Arabic, Hebrew, or Maltese) are close to Esperanto (see also Are there natural languages with the following properties (seen in Esperanto)?).