Why did the Byzantine Empire consider themselves the "Second Rome" ?

The term "Byzantine" is something made up by modern historians to call the Roman empire during the Middle Ages. In a very general sense the term "Byzantine Empire" refers to the Roman Empire after the fall of the Western Roman Empire, after the conversion to Christianity and after the adoption of the Greek language and Greek culture. The exact date when the Roman Empire could be called "Byzantine" instead of "Roman" is a hotly contested topic but most historians will agree that the Roman Empire in 400 AD was still very much Roman in a classic sense versus the Roman Empire in 1000 AD which was different in a lot of ways and therefore the term "Byzantine" is used to differentiate the two. In a sense you could think of Roman history as the founding of the city of Rome, the Roman Republic, the Roman Empire, the Byzantine Empire.

The most important thing here is that there is no break in continuity from the Roman to Byzantine periods, the people in the "Byzantine Empire" considered themselves Romans, or Romaioi (Ρωμαίων) all the way until the fall of Constantinople in 1453 AD. For example, I'm currently reading The Taktika of Nikephoros Ouranos and the term Ρωμαίων or Roman in Greek is used to refer to people of the Empire.

To give a little more background, Theodosisus I officially split the Roman Empire into two separate administrative and political halves upon his death in 395 AD. The West under Honorius, and the East under Arcadius. The Western Roman Empire (WRE) ceased to exist during the 5th century, and the last Emperor Romulus Augustulus was deposed in 476 AD. You could argue that in a practical sense the WRE ceased to exist before this date but I reference it for simplicity. On the other hand, the Eastern Roman Empire (ERE) under Arcadius went on to exist for another thousand years until the capture of Constantinople in 1453 AD. So in a sense, the Byzantine Empire didn't consider themselves "Second Rome" at all, they were THE Roman Empire and considered themselves Romans until the end.

Sources:

The Taktika by Nikephoros Ouranos

I also pulled from the course, "Late Antiquity: Crisis and Transformation" by Thomas F. X. Noble. I rather enjoyed the course and it is available through The Great Courses Company. The course starts at the split of the Roman Empire, through the fall of the WRE and on into the Middle Ages ~800 AD.