I was taught that when watching someone fingerspell you should not try to say the letters in your mind but sound them out. in other words for someone fingerspelling the word "FAST" you should sound the letters out in your head fff aaa ssss ttt instead of F....A....S....T. That way if you miss a letter you could still possibly pick up the word. Anyone please correct me if I am wrong.
Yup - that's the way that it's typically taught here too - provided the student is hearing of Hoh of course.
Another way that is often very helpful is to watch for and recognize patterns (basically the shape of the word, or part of the word).
For "patterns" you practice seeing common letter combinations as "a unit" regardless of where they are found in a word (beginning, middle, end) for example:
TH , THE, GTH, ING, ED, ORE, EIGHT, STR, ONG, LY etc.
Just like young children are taught "memory words" or "sight words" - practice learning to see small common fingerspelled words as "sight words" as well - loan signs/ lexicalized fingerspelling (BANK, BACK, OWN, WHAT,JOB, DO, VEG, SO, IF, OR,) are good "ASL equivalent" examples of English "sight words".
Also, look for the "shape changes" in fingerspelled words, just like you would in written English. Some words have a lot of shape change (APPLE, FARM, BACK, BANK) others have very little shape change (TOAST, SEE, ME,).
Basically when trying to "get" fingerspelling:
1) sounding the words out as they are being spelled (provided you have in internal aural memory, I'd can sound things out in your head)
2) learning and recognizing frequently used letter combinations (and seeing them as a "unit")
3) noticing and taking advantage of the shape changes in how letters are formed
Then it's just a matter of Practice, Practice, Practice, Practice.