Why does Kirito hide his duel swords?

Question

On floor 74, Kirito hides his duel swords from Asuna and Clyde and then uses them at the last second. Why did he hide them?

Answer 1

He hid the skill to avoid the attention and envy of other players.

Weapon skills that didn’t have clear conditions for appearing are called extra skills. They were sometimes even called random conditions. An example would be Klein’s «Katana». But «Katana» wasn’t all that rare and appeared quite often as long as you kept training the Curved Sword skill.
Most of the ten-plus extra skills that had been found up until now, «Katana» included, had at least ten people who used each of them. The only exceptions were my «Dual Blades» and one other man’s extra skill.
These two were most likely limited to only one person, so they should be called «Unique Skill». I had hidden the existence of my unique skill up until now. But from today, the news that I was the second unique skill user would spread throughout the world. There was no way I could hide it after using it in front of so many people.
...
Since then, I only trained it when there was nobody around. Even after I nearly mastered it, I rarely used it against monsters unless it was an emergency. Aside from using it to protect myself in a crisis, I simply didn’t like this kind of skill due to the attention it draws.
I even thought that it would be better if another Dual Blades user appeared-
I scratched the area around my ear and muttered.
“…if it became known that I had such a rare skill, not only would people pester me for information…it might attract other kinds of problems too…”
Klein nodded.
“Online gamers get jealous easily. I wouldn’t since I’m an understanding guy, but there sure are plenty of envious People.

As we can see later his fears were wellfounded:

Somehow they had found out about where I lived. As a result, swordsmen and information dealers had crowded around my house since early morning. I ended up having to go through the trouble of using a teleport crystal to escape.

Of note is that he wouldn't have hidden the skill if he had known how other players could unlock it.

“I’m disappointed Kirito. You didn’t even tell me that you had such an awesome skill.”
“I would have told you if I knew the conditions for its appearance. But I really couldn’t figure out how it happened.”
I answered Klein’s complaint with a shrug.
There wasn’t a shred of lie in what I said. About a year ago, I opened my skills window one day and found the name «Dual Blades» just sitting there. I really had no clue on what conditions made it appear.

The excerpts are from chapter 12 of the first light novel Aincrad (translations by Baka-Tsuki).

Answer 2

He didn't want to show that he was strong to everyone and if he showed his dual swords he would have got a lot of publicity.
He didn't want the unnecessary attention, if everyone found out he had something so unique then people would be jealous of him and start stalking. They would start questioning him about where he got the sword from.
He would make a lot of enemies because people would try to get his sword.

So he thought it would be better to just lie low and hide his power.

Answer 3

His dual swords were never hidden and Lizbeth confirms this in Sao season two part two following ggo, where this arc is in Alo and consisted of going on a last quest for the year and hoping to retrieve the weapon spotted known as Excaliber. Hence the arc title being caliber. kirito, his harem of asuna Liz silica and suguha aka leafa in the game and Klein take a quest where Lizbeth states to the entire group while blacksmithing some of their equipment, 'You said it was going to be a secret just between us but..' as she elaborates needing to use the skill that required dual blades in order to survive a boss. Kirito stated that it just showed up as a skill one day, and never using it, he wasn't aware of the skill that wasn't disclosed of what it is capable of or not hence fighting as the known dark swordsman made sense as he explains back he he wasn't aware of the sword or skill capabilities and had no experience. This is the same to say you don't pick up a rocket launcher and expect it to work when you haven't ever used one and have no experience versus multiple years of training with a style that's one sword and is classified as one of the top players recognized with his skills. Explaining out of options and everyone including kirito couldn't kill the boss he post comments about no intent on using them but realized during battle he had tried everything possible except the dual blades which is why he only used them when he had 0 options to win and not die. So they weren't hidden as Lizbeth knew all about the origins despite not explaining so in season one, and she alone knew so there wasn't a secret. His motives additionally revealed and were logical. Why fight with something you never used and don't know how the outcome is.