"I explain that I am their friend."
Translation:Wyjaśniam, że jestem ich przyjacielem.
If you used tlumaczę instead of wyjaśniam it is perfectly correct in this context. Tłumaczę = I explain/I translate. Wyjaśniam = I explain. Also in a more social context Tłumaczę się means I give an excuse/I justify myself/I defend myself etc. Hope this will help :)
There seems to be a minor typo in the text of the suggested answer given when my translation was "almost correct", though it is right as shown above. In the suggested answer in my situation it says "wyjasniam, że jestem ich kolegą": note the lack of the accent on the s in the first word (and even more minor - no capital letter).