How do the HDR textures work in the Source engine?

I peaked inside the skybox folder inside the materials and saw that they’re normal .vtf files. When I opened them up, they were a lot more “cartoony” and brighter and the colors were all messed up compared to the normal skybox textures. Aren’t HDR files .float files? Is Valve using some pseudo HDR technique?

No, they’re real HDR. 16-bit floating-point per pixel per channel.

Take a look at this: http://developer.valvesoftware.com/wiki/HDR_Skybox_Creation

But then how would I export them in a .float file? Targa is a LDR image.

You should be able to export them as .pfm, which is floating-point. You can convert from there.