Why does "Te vas a cansar" mean "You're going to get tired"? Irse means to leave, to go, to die, to go away and to forget. There is no translation which means "to become something" for instance, "to become tired".
Te vas a cansar or, alternatively, Vas a cansarte, illustrates two grammatical concepts:
Pronominal verbs, like cansarse, which means "to get tired, to become tired"; contrast this with plain cansar which means "to tire, to make somebody tired". Pronominal verbs are those which use a "reflexive" pronoun even though they're not reflexive (like cansarse, caerse, reírse, etc.).
The use of the verb ir plus a verbal infinitive to show future, as in vas a cansarte. This is similar to English to be going to. It's a very common structure because the "proper" future tense is used less and less in spoken Spanish.
Vas a + infinitive therefore means "You're going to" + whatever. And for cansarse, you have to employ a pronoun that refers back to the subject, in this case the second person singular pronoun te. Thus, vas a cansarte = "you're going to get tired". There are additional rules that say that, in this kind of structure, you can move the pronoun in certain ways; in particular, you can move it away from the back of the infinitive (vas a cansarte) and place it before the main verb (te vas a cansar). The pronoun is still bound to cansar, though.
It's because the verb cansarse means "to get/become tired." The te is a part of cansarse and not a part of irse. Context tells you which verb the te corresponds to.
Equivalently, you could say Vas a cansarte.
This phrase translates literally to "You're going to tire", but that's not idiomatic in English. So instead, it's translated to "You're going to get tired".
There are small differences in languages like this, even for simple, basic phrases. For instance "Tengo sed" means literally "I have thirst"-- but nobody talks this way in English. English speakers say instead, "I am thirsty", meaning the exact same thing, so the phrase is translated to that, instead of literally.