as for the solvent, non flammable for safety during cleaning, also to prevent flammable gasses from entering the engine after cleaning, and many solvents of that nature may damage the filter.
Oil.. thats an easy one. the stock filter is foam, and foam by itself isnt much of a filter. but add a little oil, and it starts sucking dirt outta the air very well. Oil bath filters where used for years in autos, and still are in some industrial and other applications.
And by soak, they mean put a small amount on the filter and squeeeeeeeze out about as much of it as you can get out.