A summary of the useful comments buried below:
тому is redundant here, but popular in native Russian speek.
My theory: Since назад only means "backwards", тому назад originally meant "backwards (in relation) to this (point in time)" when the starting point - from which to count back the period in question - was stated in a preceding sentence only. Like English "prior to this".
But nowadays, Russians just keep the тому for the sake of a flowing speech, even when the reference point for the back-counting is obvious - the present day in our example.
I can give one example. In Polish some people prefer (at least in the spoken language) to make their sentences longer inserting pleonasms (this applies for the written language, too), particles ("-że" added to words as a means of emphasis) and repetitions.
"I saw it yesterday." "Widziałem to wczoraj" "Ja żem to wczoraj widział" (an example sentence you may hear in a tv series parodying that kind of people, it's not just "colloquial Polish", it's bad Polish, it also presents an example of expressing the past tense in a bad way, with a weird "że+personal_ending" pseudo-auxiliary verb)
"What did you ask him about?" "O co go zapytałeś?" "O co żeś się go zapytał?" ("pytać się" isn't incorrect but I don't consider it a very literate form).
This tendency of making reflexive verbs out of transitive ones seems to be actually quite popular in casual speech (even without this evil "że+ending"), so some people would say "pytać się" instead of "pytać" or "wracać się" instead of "wracać" (again, "wracać się" is not a literate form).
Some people would overuse the word "zapytanie" ("a query") because they think that "pytanie" ("a question") is not enough. Some would use words they don't know to make their language seem more fleshed out ("tudzież" meaning "as well as" used as "or", "czy też"), ("bynajmniej" meaning "not at all" used as "at least", "przynajmniej").
To sum it up, I would say that in casual speech the language may tend to shorten, but casual doesn't necessarily mean colloquial. In colloquial language people would stack up words wanting to say more even if that wouldn't mean that what they say conveys any more meaning. And even a correct usage of some more complex structures might be preferred in colloquial speech or writing, like probably this example in here, just to preserve some kind of flow of speech.
I'm glad you bring this up. I've noticed a couple other times in this course when someone says the informal way is longer. That may appear strange at first, but in English we do this all the time. People say "Look at this here melon," where the standard version is "Look at this melon." It's not a perfect example but hopefully it conjures up some others in the mind.
In this case "тому" has the meaning "from this moment of now" / "before". And I think this sentence doesn't need it neither in English nor in Russian. Because "я родился пять лет назад" is absolutely correct. The choice with "тому" is more native. But if you want to use this one in this task you should translate "тому" in English and you would face a problem that nobody says in English this way.
"Тому" has many meanings in Russian. But in this case when it is used with periods of time and the word "назад" it is referring to any period of time of hours, days, years, etc. until now, before.
You can say this phrase without this word because it is redundant, it's an old fashioned and colloquial. This kind of expression (with this word) looks like a part of a fairy tale, but you can hear this word from your interlocutor in modern speech, so that is why this word is in this course. I do not think that this word is subject to the General rules of forming of sentence structure. It is an exception.
this might be a tad late for you, but here is a link to a page of videos using "comprehensible russian" if you go to her very first couple of videos (they're about 5 minutes long - and all free) she speaks about birth, life and death. She also says dates and stuff in REAL RUSSIAN SPEAK ... not some computer generated guff ..... https://www.youtube.com/channel/UCDNbk-uX4D6nsthi8L03fng/videos I have found these pretty useful but, like duo, like befluent, like russia club, no on site gives the full package
Interesting how final "д" on many Russian words is pronounced like a "т": https://forvo.com/word/%D0%BD%D0%B0%D0%B7%D0%B0%D0%B4/#ru
Sometimes Duo accepts incorrect spelling and sometimes it stops me when there is an obvious typo. Here I typed Я родилься пять лет тому назадь and it was accepted despite having incorrectly inserted ь in родился and in назад. This leads me to always check the comments after each question where the correct spelling is written.
That's when a computer converts text into sppech using a database of sounds. This is what the Russian course uses so that's why the pronunciation is sometimes off with the stresd on the wrong syllable or sounds mashed together. But the advantage of it is that it allows for the "turtle" function and probably also the part where our efforts saying the words can be assessed as similar enough or not... I hope that answered your question. :-)
Literally "тому" means "to that", i.e. it's the dative case of the demonstrative pronoun "то" ("that"). So it's something like "five years back to that". However, it's just an idiomatic expression and an outdated one at that, so don't pay too much attention to it. You might encounter it if you are going to read Russian literature or fairy tales, but in real speech it's rare. Normally we'd just say "я родился пять лет назад".