Lear has asked his three daughters to tell him how much they love him, promising to give the most desirable one-third of his kingdom to the one who expresses her love most convincingly. The old man is, in effect, trying to get his daughters to compete with one another in exaggerating their love. He thinks he can buy love. Both Goneril and Regan try to outdo each other in expressing the love for their father which neither of them feels. When Cordelia's turn comes to speak, she refuses to exaggerate the genuine affection she really feels for her father.
Lear really loves Cordelia too. He doesn't want to penalize her for not flattering him. When he says, ". . . mend your speech a little / Lest it mar your fortunes," he is offering her a chance to preserve her one-third share of his kingdom by expressing at least "a little" affection for him. He is also revealing that she is making him a bit ashamed of himself for trying to make his daughters look grasping and dishonest. But Cordelia refuses to lie about her true feelings. This enrages her father, and he disinherits her completely, giving her one-third of his kingdom to his two daughters to share.
Cordelia's obdurate and self-destructive behavior is hard to explain, since she really and truly loves her father, as she proves later in the play. Leo Tolstoy wrote a polemic against Shakespeare, and his "King Lear" in particular, in which the great Russian writer expresses the view that Cordelia seems to be deliberately trying to "vex" her father. Perhaps Shakespeare expected the viewer to assume that Cordelia is shamed and offended by the old man's vulgar mistreatment of his children in making them compete in describing their "love" for him in front of his entire court. She has already heard Goneril's and Regan's fantastic and implausible descriptions of their filial devotion, and seen all the passionate gestures that accompanied their speeches, and she cannot bring herself to behave like these two shameless women.
a single conversation across the table with a wise man is worth a month's study of books
mend your speech a little lest you may mar you fortunes