Most of us cast a rod with either our feet side by side or staggered slightly, with the foot opposite our casting hand forward a little. For example, a person casting a fly rod with their right hand might do so with their left foot slightly in front of their right foot, & both feet about shoulder width apart. This seems to be pretty standard feet position for casting. This is the same position you should take when shooting a rifle or a shotgun. As you cast, you put you shift your weight from the ball of one foot to the heel of the other, etc. I won't go into detail, as I'm sure you get the point & their are tons of books & videos covering the subject.
Some folks though, will benefit from switching their feet positions around, so that the foot on the same side as their casting hand is in front. Again for example, this would mean that a person casting a fly rod with their right hand would have their right foot slightly in front of their left foot & both feet about shoulder width apart. This, more than any other way, will bring your hips into the casting action. Your knees should be relaxed & slightly bent - not stiff. This will also line-up your entire body & casting hand & arm with the fly line as you cast it out away from you. This helps you to aim at your target much easier.
This method may feel odd at first. It may also sound counter-intuitive. However it does have its benefits. When you practice casting, try to do so on a lawn. Put some pie plates or other markers out on the grass at measured distances & try to hit them with your line. This will help you gauge both distance & accuracy.
If you find you're having trouble hitting your lawn targets, switch you feet around to the positions I just mentioned & see what happens. This could be a real trick to improve your casting. Hey, if doesn't work or just never gets to feeling right for you, you can always go back to your old foot positions. Then you can start examining how you hold the rod, how your arm moves, your back cast,.......
There are many aspects to casting a fly rod. Many times we make it harder than it really is. We over think the process. Sometimes all you have to do to improve is simply pay attention to your feet. Just an idea & some food for thought.