The Romans borrowed the practice of calling the planets by the name of deities from the Greeks, and thus from the Babylonians. Diodorus Siculis, a Greek writing in the first century BCE, observes:

"But above all in [astrological] importance, they [the Babylonians] say, is the study of the influence of the five stars known as planets, which they call "Interpreters" when speaking of them as a group, but if referring to them singly, the one named Cronus by the Greeks [the Roman Saturn]...they call the star of Helius, whereas the other four they designate as the stars of Ares, Aphrodite, Hermes, and Zeus, [the Roman planets Mars, Venus, Mercury, and Jupiter] as do our [that is, Greek] astrologers." (2.30.3)

The Greeks, in short, seem to have adapted the Babylonian practice of naming the planets after gods, substituting their own deities for Babylonian equivalents. The Romans simply followed suit.

Unlike the Sun and Moon, which were sometimes directly identified as gods - in the Romans' case, Apollo/Sol and Diana/Luna - the five planets were generally understood to be celestial bodies. Vitruvius, for example, describes the motion of the planet Venus in strictly mechanical terms:

"The planet Venus, as soon as she escapes from the influence of the suns' rays, runs through the space of one [Zodiac] sign in forty days; and what she loses by stopping a long time in one sign, she makes up by her quick passage through others. She completes her circuit through the heavens in four hundred and eighty-five days; by which time she has returned to the sign from whence she set out." (9.1.9)

[Edit] In late antiquity, however, the influence of Neoplatonism and a general trend toward henotheism and pantheism in pagan thought seem to have encouraged more direct identification of the planets with the gods, at least in certain intellectual circles. In the dedication of his astrological handbook, for example, the fourth-century author Firmicius Maternus calls upon:

"you, Saturn, who, set in the highest point of the sky, carries the leaden light of your planet in sluggish revolution; and you, Jupiter, dweller on the Tarpeian Rock [in Rome], who makes the universe rejoice with your benevolent power and who holds the rule of the second sphere; and you, Mars Gradivus, always with your fearful red color, whose home is the third region of the sky; and you two, Venus and·Mercury, faithful comrades of the Sun..." (1.14)

Such identifications may have been no more than literary flourishes; but they indicate that, at least for some late Romans, the association between planet and god was very close.

Plato in his Timaeus, the culmination of his life's philosophical and scientific thinking, described the creation of the universe by a supreme god, the Demiurge, as well as the composition of the heavens. He believed that the spherical earth was at the center of the universe, and the seven planets (the Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn) were the first celestial gods, created by the Demiurge out of soul-matter. Plato comes off sounding almost like an atheist at times, but he acknowledges the celestial gods must exist, since we can see them, even though we know very little about them aside from legend. However, he proposed that the Demiurge had given these gods the job of creating the other gods as well as humans and other mortal life forms, since the Demiurge was only capable of creating perfect, immortal beings. So in that "scientific" understanding of reality (Platonism), the planets really are gods.

I don't know how much of Plato's thinking filtered down to the regular citizen of Greece or Rome, but it was enormously influential on other philosophers and theologians over the following centuries.

This is a kind of related follow up question.

Is it a coincidence that Jupiter (or Zeus for the Greeks), the biggest planet, got the name of the king of the Roman gods? Or did the Romans and Greeks somehow know that Jupiter was the biggest planet?

Follow up question: it's interesting that they named the largest planet after Jupiter, the king of the gods. The planets Venus and Mars are much easier to see with the naked eye. Is there anything to this or is it just a coincidence? They didn't somehow know the planet Jupiter was the largest, did they? Is there anything else we know about why they chose certain gods for certain planets?