I don't understand why the following statement is true.
'A noncompact Lie group need not have any nontrivial tori (e.g. $\mathbb{R}^n$).
Taking $n=2$, we get $\mathbb{R}^2$. Now consider the set of all elements of modulus 1, that is, the unit circle lying in $\mathbb{R}^2$ which is a subgroup isomorphic to $U(1)$. Doesn't this correpond to a non-trivial tori?
They refer to the additive group $(\Bbb R^n,+)$. The set you describe is not a subgroup; it is not closed under addition.
By the Heine-Borel theorem, the only compact subgroup of $(\Bbb R^n,+)$ is $\{0\}$. Hence, $(\Bbb R^n,+)$ has no non-trivial tori.