I have something similar after my 3rd pregnancy...don't know why.
I tried every cream. But here's what I learned...
NOTHING will work until you get rid of the dead, dry skin. The moisturizers CANNOT penetrate layers of dry dead skin. So first, soak your feet in warm, slightly soapy water. Get yourself a pumice stone or one of those "feet graters". You want to gently remove all of that dead, flaking skin. It may be a little tender in spots where the cracks have created open sores. You still need to get all that debris away - both to help the moisturizers work and to keep infection at bay. Use a nail clippers or something like that if needbe to GENTLY cut away as much of the dead (white) skin you can around any open cuts.
Once the dead skin is removed, dry them thoroughly. The water would have penetrated the skin so it should already be plenty soft and moist. Then cover with a moisturizer of your choice - my dr recc Aquaphor since it doesn't have a lot of chemicals, scents, etc. and let's the skin breathe...they even recc this on my baby too! It's colorless and scentless so it shouldn't stain either. Put on socks and go to bed for the night so it can soak into your feet.
In the mornings, I did not remoisturize. I simply covered any cuts/sores with polysporine and a band aid to keep out any infections.
The second night I'd remoisturize and recover with socks.
The third night, I'd gently scrub my feet like the first night, removing any dead skin again.
My feet are fabulous now. But it is a constant maintenance. I don't have to always scrub and moisturize, but I am more aware of the condition of the bottoms of my feet and toes and take the necessary maintenance as needed.
As to why this happens? As I learned, we loose the "fat" in our feet over time. Age, weight gain/pregnancies cause our feet to flatten out and we loose our arches. So our feet don't really "grow", but instead get wider/longer due to flattening. This also causes our skin to react differently. There isn't as much "fat" holding in the moisture. So we have to keep the dead skin off, moisturize ourselves and keep our feet healthier as we age.
I hope this helps you!