Ooh, a chance to swing my linguistics degree around!
People learn several things when they learn to read languages with alphabets or syllabaries (where each grapheme stands for more than one sound):
1) Learning that graphemes stand for sounds, and what sounds each one stands for
2) Learning what words look like
3) Learning how words are combined in print
When you're first learning to read (at any age, although it's significantly easier for children due to neuroplasticity), you depend on the letters to tell you what the word is. Try to think back to when you were really little and trying to read, or watch a newly-reading child or adult, or think about reading in a foreign language that you're just learning -- there's a lot of pointing, a lot of staring hard at one word at a time, a lot of effort.
As you see words over and over again, you begin to memorize the appearance of the most common words. Some literacy programs specifically train these so-called "sight words" in order to speed up the learning process. You also learn what words are likely to follow other words (and in most languages this also ties in with your knowledge of the spoken language).
So, if you see a three-letter word that starts with T at the beginning of a sentence, you know that it's highly likely to be "The" -- so you won't immediately stop to read and process the whole word, you'll check to see if it's "The" and stop to read it if it's not. You also know that "the" is almost guaranteed to be followed by a noun or adjective.
However, as the examples have shown, memorizing what words look like is not a guarantee that you'll be able to read fluently -- you still have to spend more time and attention on longer and less common words. For proficient readers, this process is largely unconscious and you probably won't even be aware of the effort unless you encounter tons of unusual words, unusual usage, or words you don't know. For less proficient readers, an unusual pattern or a rare or large word can almost stop them cold.
So it's not that the letters in the middle don't matter, it's that your brain is very efficient at something you've been doing for a long time! It predicts based on probability, then once it's analyzed enough features to confirm the identity of a word, it stops analyzing that word and moves on to the next. (Again, there's some good examples in this thread.) You might've seen articles saying that we don't read straight across each line of a block of text, but that our eyes jump around the sentence or even the paragraph -- and that's for the same reason. This is why you can often get a good amount of information from "skimming."
NOTE: The learning process is significantly different for ideographic written languages, such as Chinese. I believe research there is ongoing.
