Where does the modern tradition of exchanging gifts from Christmas comes from? I've heard that it's inspired by a similar tradition the Ancient Romans had on one of their festivals, is that true?
Most recently? From the 1840s, at least in England.
Christmas has many fathers, as traditions like gift-giving and feasting have periodically risen and then been suppressed over the centuries, and then revived with newly Christian significance retroactively applied to what might have originated in a secular or pagan custom.
The most recent father of our modern idea of the Christmas season in the "Anglo-Saxon" world, is the Victorian era, with certain customs developing independently in England and in the United States. Christmas trees, Christmas cards, turkey dinners, and caroling were all popularized in the mid-19th century. This is thoroughly covered in J.A.R. Pimlott's The Englishman's Christmas: a social history (harvester, 1978).
Gift-giving was, as in many cultures, originally a New Year's Day custom; the Scots have Handsel Monday and the French the étrenne. The latter term derives from the Latin strena, a term for both the gift and the gift-giving of the Roman festival of Saturnalia, and attests to its ancient origins. In addition to gifts to each other, medieval Europeans were to offer gifts to the sovereign; the 13th century chronicler Matthew Paris writes of Henry III extorting gifts from his court.
The heathen custom was rationalized into Christianity by associating it with the gifts of the Magi described in Matthew 2 (observed on Epiphany, January 6 in the Western Chruch) and with the story of Saint Nicholas, Bishop of Myra, who was said to have given away his entire inheritance to the needy (observed on December 6). Saint Nicholas, of course, is the origin of the name "Santa Claus," and some of the associations of Santa Claus. Still, it was frequently suppressed, notably by the Puritans, and Christmas was not a particularly special holiday for centuries.
The gift-giving custom was revived in the Victorian age. It was an era where there was strong interest in reviving old customs (real and imagined), but as the popularity of Christmas grew, it displaced New Year's; Prince Albert called Christmas "a day for the exchange of presents, as marks of mutual affection and good-will." This image of Christmas was popularized by Charles Dickens in a series of Christmas season stories published from 1843 to 1848, the most famous of course being A Christmas Carol.
I'm not saying this is The Truth®, but here's the argument typically given for the Ancient Romans you mentioned.
Nobody is really sure exactly when Jesus was born (even the year, much less the exact day). The biblical authors do not seem to have felt it was particularly important information. The earliest two Gospels don't even mention Jesus' birth at all. The first time somebody deigned to come up with a date for it was two centuries later, and that date was May 20. March 21, and several different days in April have also been used. The earliest reference we have that uses December 25th is from 200 years later (400 AD). So it seems highly unlikely this date was decided upon due to certian knowledge that it is correct.
So why this particular date? Well it just happens that the Romans had a big solistice festival called Saturnalia. Of course the solistice happens a few days earlier, but Roman practice at the time was to have a week-long celebration, culminating in the Birthday of the Unconquerable Sun, on December 25. Saturnalia was a very popular festival amongst the Romans, and involved gift-giving.
So the logic for the Roman origins is that the Christians settling on the same day for the Son of God's birthday as the Roman's Birthday of the Unconquerable Son is not a really weird conincidence, but rather a purposeful attemmpt to co-opt for Christianity what was otherwise an inconveniently popular Pagan holidy. So the gift-giving came over from Saturnalia.