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why does the word gördüm mean I saw not just saw?

I'm by no means fluent in Turkish but I want to share my sight from a learners point of view.

In the Turkish language you have the personal pronouns, like you stated correctly are ben, sen, o and so on.
But on the other hand you have the pronouns suffix forms which are more important in the language.

From http://www.turkishlanguage.co.uk/personalpronouns.htm:
"(1) Stand-alone Pronouns Ben gidiyorum - I am going.
(2) Pronouns in suffix form (extended words). Gidiyorum [gidiyor-um] - I am going.
The stand-alone personal pronouns are not used widely as the person is evident from the personal verb ending. They are used for emphasis only in their simple form as the verb form itself already points to the person. This is similar to Spanish where a person will say comprendo - I understand instead of Yo comprendo - I understand."

Because of the nature of the suffix form, it already states everything which is needed regarding the person.
This is done by distinguish persons with the specific personal ending (in most cases I have to say because I'm not fluent) like in your example

görmek = Infinitive form of the verb
gör = verb stem
-di-/-ti-, -dı-/-tı-, -dü-/-tü-, -du-/-tu- = Definite Past Tense, added to the verb stem, and awaits the personal suffix to state the person

(ben) gör-dü-m
(sen) gör-dü-n
(o) gör-dü
(biz) gör-dü-k
(siz) gör-dü-nüz
(onlar) gör-dü-ler

(Source: http://www.turkishlanguage.co.uk/pasttense.htm)

The stand-alone pronouns (in brackets) are mostly not used, because you already know the person from the last part, the personal suffix form (please also note, that this is true for other tenses, but the suffixes used are different).
Therefor the stand-alone pronoun is widely omitted to support effictiveness and to avoid recursivity.

Hope that helps :)