To approach such questions it is typically crucial to recall the definitions. You do not give the definition of $\text{span}\{u_{1}, u_{2}, \cdots, u_{n}\}$ you use.
There are basically two; I assume the one which you ought to use (else you question is a bit void); see below for the other.
Definition: $\text{span}\{u_{1}, u_{2}, \cdots, u_{n}\}= \{\sum_{i=1}^n \lambda_iu_i \colon \lambda_i \in \mathbb{R} \}$.
Assuming this definition one can show that $S=\text{span}\{u_{1}, u_{2}, \cdots, u_{n}\}$ is a subspace by checking that:
- for all $v,v' \in S$ one has $v+v' \in S$.
- for all $v \in S$ and $\mu \in \mathbb{R}$ one has $\mu v \in S$.
(If you want to be very precise you'd also have to assert $S\neq \emptyset $ but that's a detail.)
For the first write $v= \sum_{i=1}^n \lambda_iu_i $ and $v'= \sum_{i=1}^n \lambda_i'u_i $ (this is possible by the definition of $S$). Then $v+ v' = \sum_{i=1}^n (\lambda_i+ \lambda_i')u_i $ is again in $S$ as $\lambda_i+\lambda_i'$ is a real number again.
The second is quite similar to check, I skip the details.
So the theorem asserts that whenever you have a set of the form $\{\sum_{i=1}^n \lambda_iu_i \colon \lambda_i \in \mathbb{R} \}$ for some vectors $u_1, \dots, u_n$ then this set is a subspace.
In fact it is the smalles subspace that contains $u_1, \dots, u_n$; one thus calls it the subspace generated by $u_1, \dots, u_n$. One could also use this as the defintion (compare a comment on the question).
Alternative definition: $\text{span}\{u_{1}, u_{2}, \cdots, u_{n}\}$ is the smallest subspace that contains $u_1, \dots, u_n$.
If you were to use that definition then the theorem would be obvious. However, you then would want to prove that $\text{span}\{u_{1}, u_{2}, \cdots, u_{n}\}= \{\sum_{i=1}^n \lambda_iu_i \colon \lambda_i \in \mathbb{R} \}$.
Either way the main thing to retain is: $\text{span}\{u_{1}, u_{2}, \cdots, u_{n}\}= \{\sum_{i=1}^n \lambda_iu_i \colon \lambda_i \in \mathbb{R} \}$ is the smallest subspace that contains $u_1, \dots, u_n$, one calls it the subspace generated by $u_1, \dots, u_n$.
