There’s very little information about how she actually came to France. There are a bunch of later legends about her that sometimes creep into popular histories, but essentially all the contemporary information comes from the chronicle of the abbey of Saint-Pierre-le-Vif in Sens, which says:

“At that time, King Henry sent Walter, Bishop of Meaux, and Wascelin of Chalons, along with others, to a certain king beyond the borders of Greece [i.e. Byzantium], who was called Yaroslav, from the land of Russia, to ask for his daughter in marriage. He sent them back to France with his daughter.” (pg. 122)

That’s all it says - it’s not even entirely clear what year it was! It was probably 1049-1050, since Anne arrived in France and married Henry I at Reims cathedral in 1051, so that would allow enough time to get there and back. But it’s also not clear whether this was one embassy, or two, which maybe took place in different years.

The journey was about 2000 kilometres, but no one really knows what route they took. Roger Hallu made some guesses about the most logical route; maybe they visited Duke Casimir I of Poland in Warsaw? Maybe Prague, and Regensburg? Maybe they visited Holy Roman Emperor Henry III in Worms or Cologne? Sometimes the various possible stops along the way are repeated as fact in non-academic histories, but there is no actual medieval evidence for any of this. Even Hallu’s imagination seems to be running wild sometimes. So we don’t know who went to Kiev, or who came back to France with her, or if anyone from Kiev stayed with her in France.

It is sometimes said that she wrote to Yaroslav complaining about the barbaric manners of the French, but that is a much later invention. Another legend is that she brought an Old Church Slavonic set of Gospels with her, apparently based on the fact that the Bibliotheque nationale in France had a copy and they didn’t know where it came from, but it’s probably much more recent than Anna’s time.

One thing she really did bring with her was some kind of jewel, or at least the evidence for it is much closer to her lifetime - Louis VI donated the jewel, which he believed had belonged to his grandmother, to St-Denis abbey.

So in summary, we know barely anything at all about her trip, except for some imaginative (but logical) guesses.

Here are some sources for Anna's life, some of which briefly deal with her journey to France:

W.V. Bogomeletz, "Anna of Kiev: An Enigmatic Capetian Queen of the Eleventh Century", in French History, vol. 19, no. 3 (2005)

Talia Zajac, “Gloriosa Regina or “Alien Queen”?: Some Reconsiderations on Anna Yaroslavna’s Queenship”, in Royal Studies Journal, vol. 3, no. 1 (2016)

Roger Hallu, Anne de Kiev (Università cattolica ucraina, 1973)

Philippe Delorme, Anne de Kiev, epouse de Henri Ier (Pygmalion, 2015)

R.H. Bautier, “Anne de Kiev, reine de France et la politique royale au XIe siecle” in Revue des Etudes Slaves, vol. 57, no. 4 (1985)

Andrew Gregorovich, Anna Yaroslavna, Queen of France (Forum, 2011)

Chronique de Saint-Pierre-le-Vif de Sens, dite de Clarius, ed. Robert-Henri Bautier et Monique Gilles (Paris: CNRS, 1979)

These sources are a bit hard to find, which is why it took a few days to answer this. A lot of them are in French, of course...there are also other sources in Ukrainian and Russian but I can’t read those!