αρέσει means something like "it is pleasing". In this sentence the "purple" is doing the pleasing and it is doing that to "my sister". The sentence could be translated more literally, crudely, word by word as:
[Στην] [αδελφή] [μου] [αρέσει] [το μοβ]
[to the] [sister] [of me] [it is pleasing] [the purple]
which you would not say in English there you would say with similar meaning: "my sister likes purple"
It does convey the same message but in an unorthodox/ungrammatical way.
It's like saying "Her, she likes purple" (Η αδελφή μου, της αρέσει το μοβ.)
There is no good reason why the object should appear first in the nominative and then in the (correct) accusative in the Greek sentence. Unless, you are talking and formulating what you are going to say, realise you're going the wrong way about it, pause and then fix it. ;)
The Romance languages have this construction, as you note with Spanish, and it goes back to Latin use of the dative with impersonal verbs, e.g., placuit Catilinae Roma exire, "Catiline decided to leave Rome," lit., "it was pleasing to Catiline to leave Rome." Here's a link to a discussion: http://www.nativlang.com/romance-languages/grammar/syntax-verb_pronoun.php There's a humorous discussion in the DL Italian about this formulation in that language: https://forum.duolingo.com/comment/3124469/Al-leone-piace-la-carne English doesn't have this construction (but cf. German das gefällt mir). Ancient Gk often used the verb ἀρέσκω impersonally in the third person sg + dative, e.g., εἰ μή σοι ἀρέσκει, "if you don't like it" (lit. if it is not pleasing to you"). The future and subjunctive forms of the verb were ἀρέσω and so you can get sentences such as εἰ ἀρέσει τῷ θεῷ, "if it is pleasing to God" (the sentence is deferential). See Smyth's Greek Grammar p. 339 for examples of Gk datives with verbs. Modern Gk doesn't have a dative; instead genitive and accusative are the indirect object markers (Horrocks, Greek: History of the Language and Its Speakers, see index under accusative & esp. p. 117; cf. Mackridge, The Modern Gk Language, pp. 56ff). In this construction, I'm guessing that it's Στην αδελφή in accusative as indirect object marker.