Claudio doesn't trust Benedick to keep his secret about his love for Hero, and...
These lines serve primarily as further exposition regarding Claudio's love for Hero and raises a question as to whom Claudio wishes to disclose his feelings. This question also raises an issue regarding the matter of trust.
Claudio doesn't trust Benedick to keep his secret about his love for Hero, and he's absolutely right not to trust him. Benedick prides himself on being able to keep a secret ("I can be secret as a dumb man, I would have you think so..."), but with very little prompting from Don Pedro, Benedick can't resist blurting out that Claudio loves Hero.
BENEDICK. ... but, on my allegiance—
mark you this—on my allegiance! he is in love. With who?
Now that is your Grace's part. Mark how short his answer is:
With Hero, Leonato's short daughter. (1.1.179-182)
After betraying Claudio's trust, Benedick brazenly challenges Claudio either to admit or deny it.
BENEDICK. Like the old tale, my lord: ‘it is not so, nor ’twas not
so; but indeed, God forbid it should be so!' (1.1.184-185)
It appears that Claudio also doesn't trust Don Pedro. Claudio thinks that Don Pedro might be trying to trick him into revealing more about his love for Hero than he wants to say.
DON PEDRO. Amen, if you love her; for the lady is very well worthy.
CLAUDIO. You speak this to fetch me in, my lord. (1.1.188-190)
This issue of trust reappears later in the play, when Don John tricks Claudio into believing that Hero has been unfaithful to him. Claudio's distrust of those closest to him leads to extremely unfortunate circumstances for himself and for Hero, the woman he professes to love.
Claudio eventually admits that he loves Hero, and Don Pedro supports Claudio's feelings.
CLAUDIO. That I love her, I feel.
DON PEDRO. That she is worthy, I know. (1.1.195-196)
As usual, Benedick has a few prideful, if wholly irrelevant and unnecessarily cynical remarks to add to the conversation.
BENEDICK. That I neither feel how she should be loved, nor
know how she should be worthy, is the opinion that fire
cannot melt out of me. I will die in it at the stake. (1.1.197-199)