They seem to mean different things (they aren't different conjugations of the
same word). Can someone explain why they use the same kanji? o.o  
And probably the exact differences between the two?
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word-choice
Please help me explain the kanji in [出]{だ}した and [出]{で}た
It seems that you know the difference of their meaning so I just don't bother
to explain either of their meaning.

It is just that Chinese do not differ meaning of words in this way. So if
you're the one to choose a Chinese word here you may have to create a new word
for one of them but that is not very like the traditional way.

By the way, Chinese use preposition to differ these two kinds of meaning.

So to be more specific, check <http://kanji.jitenon.jp/sp/kanji/035.html> ,
you can see that both 出す(だした) and 出る(出た) are there.
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The both use the same kanji because they are the same in **essence** , the
meaning will be something roughly like **Something/someone left a place**.

  * **出す** is a transitive verb which means **to take out(something, from the place it was)**. e.g. 引き出しから眼鏡を出して下さい。 Please take out the glasses from the drawer.
  * **出る** is an intransitive verb which means **to leave(someplace, where the person was)**. e.g. 夜に成ると月が出ました。The moon came out when the night arrived.
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出す(dasu): to take something out.  
出る(deru): to go out of someplace.  

私はゴミを出した。watashi wa gomi wo dashita.  
I took the garbage out.  

私は家を出た。watashi wa ie wo deta.  
I left the house.
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出す（だす;da-su）means to take out, or to put out.

出る（でる;de-ru）means to go out, to leave, etc.

They both share the same Kanji root because the meaning of the kanji is
directly related to both of these verbs; both dasu and deru have something to
do with "out," going out, putting out, etc.

Usually these two verbs are taught together as transitive and intransitive
pairs.

Transitive verbs take a direct object.

Intransitive verbs don't take a direct object.

You can usually tell the difference between them with a simple test: ask what
you are verb-ing. if it makes sense then it's a transitive verb.

EX: You eat what? I eat an apple.

Here apple is the direct object. The verb takes a direct object, therefore it
is a transitive verb.

They are two different words, as _go out_ and _take out_ mean different things
in English too. So if it's easier to learn them as separate and complete
words, rather than as transitive/intransitive pairs, then go for it.
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