I want to Prove the following Theorem.
Theorem : Trace-zero functions in $W^{1,p}$.(Evans 259p). Assume $U$ is bounded and $\partial U$ is $C^1$. Suppose that $u\in W^{1,p}$. Then
$$u\in W^{1,p}_0\Leftrightarrow Tu=0\text{ on }\partial U$$
The proof of the ($\Leftarrow$) direction use partitions of unity and the flattening of $\partial U$. So one can assume $$U=\mathbb{R}_n^+=\mathbb{R}^{n-1}\times \mathbb{R}_+.$$ and $$ u\in W^{1,p}_0,~u\text{ has compact support in }\bar{\mathbb{R}}_+^n$$ $$Tu=0\text{ on }\partial\mathbb{R}_+^n=\mathbb{R}^{n-1}.$$
I understand all calculations in the proof but I can't understand the assumption. I want to know why I can make such an assumption.
It seems that your problem is with partition of unity, i.e. how a global statement is prooved on small pieces.
Assuming $Tu= 0\ $ on $\partial U$, one wants to show $u \in W_0^{1,p}(U)$. This space is defined a few pages before (p.259 in the 2010 edition) as the closure of $C^{\infty}_c(U)$, i.e. $\exists\, (u_m)_{m\in \mathbb{N}},\ u_m\in C^{\infty}_c(U)$ such that $$\lim_{m\rightarrow \infty} \lVert u_m - u \rVert_{W^{1,p}(U)}=0 \tag{1}$$
Having a partition of unity allows us to write something of the form (cf. a few pages before § 5.3.2, Thm 2 p.265 in the 2010 edition, "global approximation by smooth functions) $$ u \equiv 1 \times u \equiv \sum_{i=0}^{+\infty} \zeta_i u \tag{2}$$ (I write $\equiv$ to mean equality as a function, i.e. equality at each point. The sum p.265 is subordinated to some open cover $\{V_i\}_{i\in \mathbb{N}}$ but in our case it can even be chosen finite by compactness of $\partial U$) with a condition on Support$(\zeta_i u)$.
The calculations (that you understand...) show $$\lim_{m\rightarrow \infty} \lVert \zeta_i (u_m - u) \rVert_{W^{1,p}(U)}=0 \tag{1}$$
Either one has a finite sum in (2) and one obtains (1), either the sum is infinite and one needs a stronger condition, example adapted from p.265-266 (2010 edition), $\forall\ \epsilon > 0,\ \exists N_i \in \mathbb{N}$ such that $$\lVert \zeta_i (u_p - u) \rVert_{W^{1,p}(U)} \leq \frac{\epsilon}{2^{i+1}},\quad \forall\ p\geq N_i$$ to obtain (1).
One cannot skip the use of partition of unity because the very definition of "$\partial U$ is $C^1$" is formulated locally.