There are many secrets that exist within the elven realm of Lothlorien, including Galadriel’s magic mirror which allows her to see events of the past and the future, her elven ring of power Nenya, and the long-kept knowledge of Lembas bread and it’s sustaining properties.
But when the members of the fellowship arrive at the borders of the golden woods, they are made to wear blindfolds as they are taken to the center to meet the lord and lady who are the rulers of the kingdom. When they try to protest, especially Legolas, who is also of elven kind, and kin to the people of the woods, they are told that it is the law of the land. But why was this law put in place?
First, to clarify, it is only Gimli in the beginning who is requested to wear the covering. There has long been a history of distrust and dislike between the elves and the dwarves, and although he starts off with insulting the people of Lothlorien (who insult him right back), he argues obstinately that if he is to be treated in such an unfriendly manner, and not as a friend of the fellowship, picked for his loyalty and trustworthiness, then he would rather not venture into the lands at all.
Aragorn, who is currently trying to lead and keep order within the companions after they have just lost Gandalf in the Mines of Moria, states that if Gimli must go forward blindfolded, then they all will. He wants to create a show of solidarity within the fellowship, and the others readily agree. Aragorn too is a friend of the elves of this particular realm, as it is the birth-place of Celerian, the mother of his true love Arwen, and he has visited in the woodland before, so the elves are unsure of binding him, but as it is the only way to allow the dwarf into their land, according to the law, they accept and cover the eyes of everyone.
It isn’t specified exactly how long this law has been in place, but many Lord of the Rings fans believe that it is a recent addition to the scripture, after the dwarves, with the aid of Bilbo Baggins, escaped the elven prison of Mirkwood, 60 years prior, during their own quest to retake the Lonely Mountain and their homeland. Bilbo, with the use of the One Ring of power and its enchantment of invisibility, was able to free his dwarven companions and they managed to escape down the river in some large wine barrels, much to the chagrin of their captors. News of their escape must have spread, and it therefore makes sense that the other elven realms would wish to tighten the security of their borders, especially against dwarves.
However, it could also have to do with the increase of Orcs and dark creatures around the lands. Mirkwood has been re-infested by the ancient spiders who settled there, and also suffered the loss of yet another prisoner when Gollum, who was in their keeping, managed to escape up a tree, as a band of Orcs ambushed his guards down below. The violence and foul play were becoming more and more common at the time when the fellowship arrived, and the elves of Lorien had begun noticing shifty behavior from all manner of creatures on Middle Earth, so they could not be too careful when it came to the dwarf Gimli. But, when they get close enough into the center that Lady Galadriel can read their thoughts and hearts, she commands them all to be released from their blinkers, as they are trusted allies and enemies of the dark lord.
Many conversations, feasts, and nights rest take place in Lothlorien, where the fellowship is allowed to regain their strength. The elves sing a lament to the fallen Gandalf, who will later come back as Gandalf the White, and they bestow precious gifts upon the comrades before pointing them on the safest route to continue their journey. The gift Gimli asks for is a strand of the beautiful Galadriel's golden hair, and she gives him 3, on the condition that he not think ill of them for their caution. She says ‘May it be a sign that though the world is now dark, better days are at hand, and that friendship shall be renewed between our peoples.’
From this point onwards, there is, in fact, a change, because Gimli and Legolas do grow closer, growing to see each other more as equals, and challenging each other to many games, including the drinking celebrations after Helm's Deep, and who has the highest kill count of the Haradrim at the battle of Pelennor Fields. The acceptance that is shown towards the other races, and the happiness that can be found within seeing the good in people rather than assuming the worst, is a key theme throughout the books that makes them relatable and popular to this day.