He didn't want anyone to see that the scarf belonged to the married woman he was having an affair with.
Explanation:
Aramis received the handkerchief from Madame de Bois-Tracy, a married woman, with whom he had great admiration and with whom he flirted. The flirtation between him and the woman was reciprocated, but she was married and one of Aramis's companions was a friend of her husband, so when he dropped his handkerchief, Aramis tried to hide it with his foot, which only increased curiosity of his friends.
