How to replace fan fixture for a flat LED light fixture

The power (from the breaker) is likely at the light fixture. When you hook black to black and white to white, the light is powered directly from the breaker. In this configuration, when you flip the switch on you create a short through the switch.

What you need to do is hook it up like this.

Notice the white wire that runs between the switch and the light has a black stripe on it (in above image), that lets anybody that works on this light in the future know that the wire is switched. You can mark the wire using a bit of electrical tape, or a marker.

Figuring out which wires come from the breaker, and which go to the switch will require some tools. First Turn off the breaker, and verify the power is off using a non-contact voltage tester. Now pick one set of wires to work with first (one black and one white from the same cable). Turn the switch to the on position, and use a multimeter to check continuity between the two wires. If the wires are connected (low resistance), these are the switched wires. If not (infinite resistance), check the other set of wires. If neither pair pass the test, contact an electrician.

If at any time you don't feel comfortable doing this work, or you don't have the proper tools to complete the task. Contact a local licensed electrician.

There is a junction box somewhere at the end of the cable, best to find it and start your wiring from there.
You can cut the cable, and use the wires, but it is difficult without the right tool, and then you have to hope you have enough length and use the proper fittings to mount it to the new ceiling box.