The lines, that you see are "normals" and shows radius, twisting and direction of curve.
You can do following:
1) change radius. For example if you press Ctrl+A in object mode(radius multiplied by object scale), when A(to select all), W -> Set curve radius in edit mode, it will reset radius to 1. Or change radius of selected points in N-panel:
2) Also you can turn off displaying normals, or make them small. You can find this settings in N-panel (in edit mode):
3) Also, 2D-curve doesn't show normals. You can switch the type in curve settings