I am wondering why do we say "It's about time that I did something about that.", because I was expecting the present tense after "It's" and here I don't get why we say "I did".
This is what is known as the subjunctive mood.
Subjunctive moods occur when something is doubtful, desirable, wishful, etcetera. Unfortunately, English has no rules for the subjunctive mood, nothing really regulated, that is.
You state in your question, that you would expect there to be present tense. You are also correct in this assumption, but because subjunctive moods do exist in the language, we have the OPTION to use it.
It's about time that I did something....
Typically, subjunctive moods will occur after the word that. To create a subjunctive "tense", we take the third person past tense and use it instead of the present tense.
It's about time that I do something
This sentence is very direct. It is a complete statement. It has the same kind of context as saying
I am going to do something.
It is as if you were stating something that WILL occur in the future.
Now, like I said earlier, if there is any possibility that the event might NOT occur, we use the subjunctive.
It's about time that I did something.
Third person plural (past tense) of
to do = they did.