Prove that the range is a subspace In Linear Algebra Done Right, it said 

If $T \in \mathcal{L}(V,W)$, then range $T$ is a subspace of $W$.

Proof:

Suppose $T \in \mathcal{L}(V,W)$. Then $T(0) = 0$, which implies that $0 \in \text{range } T$.

If $w_1,w_2 \in \text{range } T$, then there exist $v_1,v_2 \in V$ such that $Tv_1 = w_1$ and $Tv_2 = w_2$. Thus
$$T(v_1+v_2) = Tv_1 + Tv_2 = w_1 + w_2$$
Hence $w_1+w_2 \in \text{range } T$ 

Why $w_1+w_2 \in \text{range } T$ ? I can follow the steps but don't understand why it is in the range

Similarly, I don't understand why $\lambda w$ is in the range of $T$
Range of $T$ consists all vectors of the form $T(v)$ where $v \in V$. $w_1+w_2=T(v_1+v_2)$. Hence $w_1+w_2$ belongs to range of $T$. If $w=T(v)$ then $\lambda w=T(\lambda v)$ so $\lambda w$ is in the range of $T$.
Note the previous line: $T(\text{something}) = w_1 + w_2$. Thus, there is something that, when we hit it with $t$, we get w_1 + w_2. The proof for $\lambda w$ will be similar: since $T$ is linear, $T(\lambda v) = \lambda T(v) = \lambda w$, so there is something that, when we hit it with $T$, we get $\lambda w$, so $\lambda w$ is in the range of $T$.