The Russians imported a great deal of cultural and political institutions from the Byzantine Empire, due to the fact that for a great deal of time, Constantinople was the primary trading destination for most of the Rus' states (especially Kiev, the most important of the Rus' states during the early Middle Ages).
Political ideals such as Caesaropapism (allowing a secular authority such as the Tsar wield supreme ecclesiastical authority), the idea of governance by a bureaucratic aristocracy, and the absolutism of the Russian Tsar, were all imported from the Byzantine Empire. Indeed, even today, one can see remnant traces of Byzantine political process in the modern Russian state.
Furthermore, the Russians imported religion from Constantinople. After the East-West Schism, the Russians remained Orthodox Christians, and maintained the religious heritage promulgated by the Byzantine Empire.
The "Third Rome" theory is not new - it developed almost as soon as Constantinople fell. The Russian Princes legitimately saw themselves as inheritors of the rich cultural and political heritage of the Byzantine Empire - even going so far as to marry disenfranchised Byzantine princesses. As a rejection of the decay of Orthodoxy in the West, the Rus' states attempted to closely mimic the deceased Byzantine Empire. Interestingly, the original sentiments for the "Third Rome" referred to the whole of Russia in a metaphorical sense as a bastion of Orthodoxy and Byzantine political tradition, then to Moscow in a literal sense as the successor to Rome as the capital of the Roman/Byzantine Empire.
I've done some study into the 'Third Rome' following the Council of Florence in 1439 prior to the fall of Constantinople in 1453.
The Russian's saw the Council of Florence as a betrayal to their religion. Unlike some Ukrainian churches, and I believe (although not certain) some Bulgarian churches, the Russian church did not want to be included in this heretical council and from 1439 you start to see Russia move away from Byzantine church.
The Metropolitan responsible for representing Russia at Florence, Isidore, was considered a traitor to Russia and was essentially banished from Moscow. In the place of Isidore we see the rise of a Metropolitan called Iona (for some reason you will find this anglicised and he is referred to as Jonas but if you read the Cyrillic it is clearly Iona). Under Iona Russia de-facto (not de-jure) break away from Constantinople in a religious sense and we see the establishment of the 'All Russian Church'. I don't think the break away is formally recognised until quite some time after.
So now we can understand the context a little better as to why Russia had broken away from the Byzantine Empire. They were essentially irritated by the Council of Florence which created the 'Uniate' church (which was some bizarre mix of Orthodoxy and Catholicism that was seen as heresy by the Russians) and formed a new Church.
Now for the 'Third Rome' theory. This idea was developed during the reign of the mighty Ivan III or Ivan the Great (as I like to call him because I thought he was swell). Ivan the Great expanded the Muscovite rule in Northern Rus' (helping quell the naughty Novgorodians) as well as smashing up those irritating Tverians. He also, very importantly, married Sophia Paleologue who was the niece of the last Byzantine emperor. By doing so Muscovy (I think during this time frame we should refer to 'Russia' as Muscovy rather than Russia as that's is how contemporaries would have referred to the region) was seen by some as a 'sucessor' to the Byzantine emperor (in a political and cultural sense).
So it was during Ivan III reign that the monk Filofei stated that 'two Romes have fallen' and a third stands! Now at the time Ivan the III was not going around saying Muscovy was a third Rome, the theory was pretty unknown by many and we see almost no references to the Third Rome theory in the 15th century.
It was not until the 17th century during another schism in the Russian church (due to the reforms of Patriarch Nikon) that the Third Rome theory really starts to ring true with a lot of Russians (known as the Old Believers). So the theory in some sense was irrelevant during its time, but grew in popularity centuries after it was first said.
Lastly (I'm sorry about all of this, I just get a bit carried away when someone asks questions about shit I actually know about) - Are there any legitimate claims that Muscovy was a successor to the Byzantine empire?
Yes! Because Ivan the III did marry the lovely Sophia Paleologue so in a purely dynastical way Muscovy was in some senses a sucessor.
However, they didn't really hold the same influence as Byzantine did during its heyday, nor (at the time) did they really have the multi-cultural, multi-lingual empire that the Byzantines had. In a lot of sense Muscovy was not a legitimate sucessor and one could argue they were more the successors of the 'Kievan Rus' state which was destroyed by the Mongols.
I hope some of this has helped you! Please ask more questions if you want me to clarify anything.
I'll have an edit in a bit to make this all a bit more succinct.