The double-headed eagle, looking both East and West, was a symbol of the last ruling dynasty of Byzantium, the Palaiologi, and represented their authority over both secular and spiritual matters. One of the last heiresses, Zoe Palaiologina, was the niece of the last ruling Emperor, Constantine XI. When the Ottoman Turks sacked and occupied Constantinople in 1453, Zoe's father Thomas was recognized as the last heir to Byzantium by the Pope, but his own lands were invaded by the Ottomans soon thereafter, and Zoe ended up in Rome as a ward of the Pope in 1465, the same year her father died. Here, she was probably Christened into Roman Catholicism, and changed her name to Sophia ("Wisdom").

In 1469, the Pope began arranging her marriage with Ivan III of Russia, also known as "The Great", after the death of his first wife. Upon his marriage to Sophia in 1472, Ivan began using the Byzantine double-headed eagle as his seal. The Papacy hoped that this marriage would re-unite the Eastern and Western Churches (under the authority of the Roman Pope), because after the Muslim Ottomans took control of Constantinople, Moscow assumed the role as premier city of the Eastern faith. The idea of Moscow as "The Third Rome" and proper heir to Constantinople as the heart of Eastern Christianity dates from this time. Anyway, the hope that the marriage would re-unite the Church was crushed when Sophia returned to the faith of her childhood. She went on to become an imposing political force in her husband's court, and had a fascinating life, but that's how the Double-headed Eagle came to Russia. It went on to be used by the Romanovs, and so was completely removed from the symbolism of the Soviet Union, but upon the creation of the Russian Federation, the old symbol was resurrected as a national one.

The other image you posted is actually not the coat of arms of the House of Habsburg (which is a huge family, with many branches, each with their own coat of arms that have varied thru the ages), but rather the coat of arms of the Holy Roman Emperors (the Austrian Habsburgs and the House of Habsburg-Lorraine.) As for its use by the Holy Roman Empire, they too wanted to associate themselves with the power of Rome and Byzantium (as well as Russia, a major power in the region), and adopting the same heraldic symbols was one was to do that. The Double-Headed Eagle appears in a number of Germanic rulers' coats of arms, from Charlemagne onwards. Sometimes it is a single-headed eagle, but the German version is generally a black bird on a yellow field, as opposed to the golden bird on a red field that is more typical of Russian versions.

The scepter and globe clutched in the eagle's claws symbolize royal authority. The scepter is an ancient symbol of authority, found in many cultures back to ancient Egypt and Mesopotamia, and probably even further than that. The globe is known as a globus cruciger, "orb and cross". It is a Christianized version of the orbus terrarum, a globe representing the earth, which was often included in statues and images of gods and Roman emperors, symbolizing their dominion over the earth. The cross adds symbolism of the sovereign's submission to the authority of Jesus, and the dominion of Jesus over the world.