The theory and design of antennas is a specialized field, with entire books and hundreds of scientific papers devoted to it. With that understood, let's focus on this particular style antenna, and ways to design it.
Blade antennas like this can be a good choice for use on a rocket, aircraft, ship, or vehicle fuselage. The solid one-piece design, with the radiating element part of the grounded mounting plane, gives it mechanical sturdiness (withstands high g forces). Being at DC ground prevents static electricity buildup (from rapidly moving air) and helps shunt lightning strokes away from the RF circuitry.
Begin by thinking of a 1/4 wave vertical monopole antenna. Because most of the radiation comes from the lowest, high-current part of the antenna, you can fold the top sideways, maintaining resonance, and not losing much efficiency. Make it fat, for broader bandwidth and higher power handling capacity. Taper the front for aerodynamic reasons.
Then you just have to understand that while you're most familiar with series-fed monopoles -- a whip antenna insulated from ground and connected to a coaxial cable center conductor -- they can alternatively be shunt-fed, by grounding the base and connecting the feed to a point partway up the antenna. By moving the feedpoint up and down the monopole (sideways in this case, since it's bent) the antenna's feed impedance can be varied.
Now, how to actually design?
You can use an electromagnetic field simulation software package, and adjust the parameters (blade length, cross-section, spacing, feedpoint etc) until you get the performance you require.
If you can't use simulation, you can build low-cost test versions and measure their impedance characteristics over the desired bandwidth with a VNA. Because of the skin effect, you can build models by folding thin copper sheet into the proposed shape at a lower cost than milling a block of solid metal. Tin snips, conductive copper tape, and solder can be your friends for quick adjustments. And because of electromagnetic scalability, you can construct test models much smaller or bigger than the final antenna and still get reasonable test results.