Why use 了 in saying I'm hungry? Why do you use 了 after 我饿？ I thought it was used to denote completion or the past tense of something, but isn't being hungry present tense?
Generally speaking, 了 following a verb indicates completion, while 了 at the end of a sentence or phrase indicates a change in state (and sometimes other things, 了 is very complicated).  A common use of this final 了 is to draw the attention of the listener to the fact that something is now the case (that wasn't previously).  In this case 我饿了 is saying: "I'm hungry now," and drawing the listeners attention to the new hunger.  Of course, when the verb is at the end of the sentence the use of 了 can be ambiguous.  Such an 了 could be used in either, or even both, of these ways, but usually context makes the intention clear.

More general information on 了 can be found here: http://resources.allsetlearning.com/chinese/grammar/Uses_of_%22le%22