As Tea Drinker said, a "part" is some arbitrary unit based on whatever you're using to measure: a bucket, scoop, etc. (I used a 5 gal pail as my "part" when I'm mixing potting soil.)
To figure out how to make X amount of total soil:
First add up the parts: 5+4+3+2+1+0.5+0.25 = 15.75 then round to make it even (makes the rest of the math easy). Your recipe has 16 parts. (Potting soil recipes don't really call for exacting precision.)
Take the total amount you need, and divide by the number of parts. In your case, this is trivial: you need 16 cubic feet, so each part should be 1 cubic foot (or about 7.5 gallons).
If you need to go out and buy raw materials, you'll need to multiply the parts by the measure you determined to know how much to buy. For example, you need 5 parts of compost at 1 cu ft/part, so you need to get 5 cu ft of compost (or get 5 x 7.5 = 37.5 gallons, about 8 5gal pails).
If you're filling a raised box, I'd make more than you think you need -- the soil will settle pretty quickly over time, and when you're watering, with rain, etc.
As an aside, Google is helpful at converting units. If you search for "1 cu ft in gal", it will give you the answer: 1 (cu foot) = 7.48051948 US gallons.