You have to learn to trust him at least until he gives you a reason not to and this isnt it.
His reason is he doesnt want you to worry, but the other alternative is he knows what a pain it will be to explain to you hes talking to is female friend and so hes saved himself the aggro.. It may be some of the conversations you wouldnt approve of (flirting), but you should give him his own space.
If you are going to be on his back, stressing and worrying about it, then you do yourself and him a disservice.
Would it be better if he left the messages and then sent the convos for you to approve? Going down that route will split you.