Without the Old Testament, the New has no foundation. The Old tells about creation, the original state of man, the creation and fall of satan, the entry of sin into the world (and thus the need for a Savior), God's desire to have a people for Himself and the character of God as revealed through the Law. The New Testament simply assumes all these, and builds upon them.
Take away the Old Testament, and you have no explanation of how the world came to be (leaving people free to believe in evlolution), no explanation of satan (leaving people free to think he is "equal but opposite" to God), no explanation of sin (so maybe we're really all good people underneath) no explanation of why we need a Savior (to save us from what?) God did not need to re-iterate these things, because they were already revealed in the Old Testament, but if you take away the OT, many things in the NT don't make sense. That's why the NT writers frequently refer back to the OT to explain and support their teachings.
blessings,
Lynn