Translation:She assumed that I did not speak Swedish.
In English the word 'didn't' is in the past tense. Which is the reason why the word 'speak' is not.
This is because in indo-european languages the first verb of a clause will contain the tense and the second verb of a clause will (almost) always be in infinitive form.
This sentence is composed of a clause (hon antog att) and a subclause (jag inte talade svenska).
Swedish does not use the verb 'to do' for negative simple past forms like english does. The verb of the subclause (talade) is in first place and is therefore kept it in simple past form.