The short answer -- NaOH has to dissolve and become a liquid solution before it can be used to make soap. So if you think any liquid will work to dissolve NaOH, then try it and find out -- does dry NaOH dissolve in oil? (It won't.) So if you mixed just fat and dry NaOH, you would be waiting a very long time to get your soap made.
The longer answer -- Water is a "polar" molecule, meaning it has a distinct negatively charged end and a distinctly positively charged end. NaOH is also a strongly polar molecule. Because both of these molecules are polar, water is able to dissolve solid NaOH. Other polar solvents besides water can be used to dissolve NaOH -- glycerin being an example.
When NaOH dissolves in water (or other appropriate solvent) it dissociates, meaning it breaks apart into Na+ ions and OH- ions. It is the Na+ ions that actually do the job of saponification, not the NaOH molecule. Water helps this dissociation process along, because it too can dissociate into H+ and OH- ions. Glycerin, despite being a polar molecule, doesn't dissociate as easily as water or NaOH, so one has to use heat and time to encourage the NaOH to dissolve in glycerin.
Fats are typically non-polar molecules, meaning a fat molecule doesn't have any region in its structure with a strong positive or negative charge. Non-polar molecules can dissolve other non-polar molecules, but they don't dissolve polar molecules.
