So basically if you need a very simple straight cut all the way through an object in one direction you can just draw in a 2D curve and use the curve directly in the boolean.

But when you want something that's more customized than that like here you want something that's not just a straight-direction extruded cut and also it's limited to a certain area and not all the way through, then you need to build a surface or solid for the cutting object instead of just a curve.

You don't have to have a solid for the cutter - a surface that fully crosses the solid and so divides it can also be used as a cutting object in the booleans.

In order to construct a custom swoopy surface you'll usually be using Sweep or Network and they want to be able to identify the individual cross-section curves so you will usually give them individual curves and not just a single all-together connected piece like you had before. But if your piece is all planar then it is ok to make it all one joined piece and then in that case you can use Construct > Planar to build a surface through a planar closed shape like that.

- Michael