I believe in trusting my intuition. 
I believe children are our future.
I believe Jesus will come again.
I believe humanity won’t survive the century.
I believe sex is beautiful and natural.
I believe myth is more potent than history.
I believe I can do anything I set my mind to.
I believe everyone deserves a second chance.

Why say "I believe X" instead of just saying X?  After all, we typically claim to believe most what we say.  Sometimes "I believe X" indicates you are especially tentative and open to persuasion about X, but that doesn’t seem to cover the above examples, nor the famous "This I believe" essays, on authors’ "rules they live by, the things they have found to be the basic values in their lives" and "the core values and beliefs that guide their daily lives."

These examples seem to be 1) clear value statements, 2) obvious truths few would dispute, which seem to represent values, and 3) controversial factual claims.  They all seem to indicate a strong emotional attachment, which might be can be fine for values, but is a rationality no-no for factual beliefs, especially controversial ones. 

If you feel tempted to say "I believe X" and can feel your emotions swell with the evil pleasure of attachment via belief, watch out!  Beware that road to rationality ruin. 

Added:  People rarely use "I value X" as a roundabout way to express a factual belief.  So their frequently saying "I believe X" as a way to express values seems to me further evidence that people often see values not as irreducible differing preferences, but as conditional values that we would share were it not for differing fact-like beliefs.  That is, we can imagine possible worlds in which the other values would make sense, but we believe we are not in those worlds.