looks like a "table" to me...
not a desk, but whatever it is, the leg construction is simple and modular..... make one and the other 3 are the same. They are independent of one another, or so it seems from the photo.
I see a plywood plate, probably about 7" wide and 20" long, which is first attached to the legs, by screwing downward from the top into the legs and braces. then simply screw the plate to the bottom of the desk/table and you've done it, by golly.
Since it's painted the screw holes into the braces can be filled and painted over. Otherwise, a more sophisticated joint would have to be used, if not a paint finish.... mortise and tenons or dowels.
The answer to your question will only be as detailed and specific as the question is detailed and specific. Good questions also include a sketch or a photo that illustrates your issue. (:< D)