I do not understand the part highlighted by blue. My trial stopped at $Tv=a_1Tv_1+\cdots+a_nT_n.$
I think my problem is that why we need to prove $rangeT \subset span(Tv_1, \cdots, Tv_n)$, and $span(Tv_1, \cdots, Tv_n) \subset range T$.
I see this pattern quite often in Proof. Can you explain it?
And I do not understand the last part why $Tv_1,...,Tv_n$ are contained in range$T$?
In order to prove that a set $A$ is equal to a set $B$, you need to prove that every member of $A$ is also a member of $B$, which is exactly the same as proving that $A\subset B$, but you also need to prove that every member of $B$ is also a member of $A$, which is exactly the same as $B\subset A$. This is why proving that two sets are equal to each other boils down to proving that each one of them is a subset of the other. In your case $A$ is the range of $T$ and $B$ is the span of the vectors $Tv_1,\dots Tv_n$.
By definition, the range of $T$ is the set of all vectors of the form $Tv$ where $v$ is a vector in $V$. Therefore, the vectors $Tv_1,\dots Tv_n$ are all in the range of $T$.