Can anybody hear explain to me how Romanian became to be considered a Romance language? I totally understand that the area was at one point a part of the Empire, but wasn’t Greek later the Preferred language of the east?
Not to mention the heavy Slavic and Ottoman influences of later eras, what caused the isolation of that area allowing Romanian as a derivative of Latin?
It's not as isolated as you think. Romanian is a dialect continuum ranging from Istrian/Istro-Romanian in Slovenia and Croatia (which are adjacent to Venice) to Moldova in the east and south to Greece (Aromanian). It is also a point of contention where the proto-Romanian homeland exactly was, but the immigrationist theory has Slavs living in modern Romania until some time following the 5th century CE, and Romanian developing further west. So it is very likely Romanian speakers were contiguous to other romance speakers at the time of its development - not that any of this can change the language's geneology. Romanian speakers outside Romania, called Vlachs, were partially assimilated by Greek and South-Slavic communities. Dalmatian was a romance language unrelated to Romanian occurring geographically between Italy and modern Romania.
Romanian langauges were definitely influenced by non-Romance speakers. The Romanian in Romania (also called Daco-Romanian) contains about 5% originally Slavic roots. The Romanian in Moldova contains up to 20%, with additional Ukrainian loans. Aromanian has been influenced by Greek in its phonology, morphology, and syntax. It contains spirants found in Greek but not Daco-Romanian. It forms the pluperfect synthetically (that is, with an auxiliary like in English), and has lost pure infinitives - this last one is a later areal feature common to Balkan langauges, what we call the Balkan Sprachbund, so probably not an original formation. Daco-Romanian also contains roots from Greek, Hungarian, German, and some of unknown origin believed to have Dacian or Thracian analogues. 90% of its roots are still Italic, derived from Vulgar Latin (although some of these are also later loans from French and Italian).
Regardless of all of this, the language is unconditonally a Romance language. Percentage of roots isn't even a good rubric for determining language geneology. Over 65% of English roots are non-native, with 60% coming from Greek and Latin - but English is geneologically Germanic. What matters is the structure, not the lexicon, and Romanian preserves the structure of Vulgar Latin even more conservatively than other romance languages like Italian or Portuguese. For example it preserves the Latin declensional/case system, except for the ablative. It also, in a way, preserves neuter gender nominals.
We understand very well the hundreds of sound changes required to transpose the Latin vulgate to modern Romanian. For example, the loss of word-final consonants, or stressed [e] diphthongizing to [je]: Lat. pellum > Rom. piele. Romanian definite articles are derived from Latin demonstratives, but they are tellingly enclitic (come at the end of a word), which is another feature of the Balkan Sprachbund.
Just as there are still places where dialects of Greek survive in Italy (see the Griko language) there are some villages in Greece where they also still speak Romance languages more similar to Romanian than anything else in that language family (see the Meglenitic language and the Aromanian language) and there are similar islands of what you could call Romanian dialects in every other nation on the Balkan peninsula (see, for instance, the Istro-Romanian language). The common features of these dialects were already attested in Vulgar Latin as it was used across the Balkanese peninsula before it was replaced by Slavic^1 as the lingua franca and it is represented in Latinate placenames across the whole peninsula as well ^2 , which can lead us to conclude that what survives in Romania has developed from the Latin language as it was spoken in the Eastern Empire on the entire Balkans. As to why it has survived so well of all places in Romania (which has only been Roman for 170 years!), this has indeed caused tons of puzzlement and wild presumptions and debates and has been called an enigma and a miracle.^3 There was a standpoint that the Roman language was dead in Romania after Aurelian evacuated Roman troops in 271 C.E. and that it was brought back by refugees from the Adriatic coast fleeing from the incoming Slavs. This, however, has been called into question by the discovery of Latin inscriptions from the fourth century AD.^2 The Romans, in other words, left, but, to some degree, their language remained in use and changed together with the Vulgar Latin language spoken across the Roman border (and has, for some reason, survived it's decline in the Empire. Go figure!).
Sextil Puşcariu, "Zur Rekonstruktion des Urrumänischen" (Festschrift Meyer-Lübke, I, pp.17-75, Halle-a.-S., 1910)
Grigore Nandris, "The Development and Structure of Rumanian" (The Slavonic and East European Review Vol. 30, No. 74, Dec., pp. 7-39, London, 1951)
G. I. Brătianu, "Une énigma et un miracle historique : le peuple roumain", (Bucarest, 1937)
Focusing on language alone, Romanian is a romance language because it is a direct descendant of vulgar latin, and later influence doesn't change that. You can't change the language family a language belongs to by external influence, loanwords are usually discarded when trying to establish genetic relationship between languages.