Mitchell Spector explained the basic ideas and the completeness (+soundness) theorem. But let me give you some more in-depth explanation of the model-theoretic perspective.
In model theory, usually${}^\dagger$, we only really care about semantics. I believe that for most model theorists, the true meaning of the completeness theorem is that we can use the familiar rules of inference to derive semantically true statements from other semantically true statements. This is (as far as I know) in contrast to non-first order logics, where we may have no (complete) rules of inference.
Consider the formulas $\varphi_1(x)= 0\leq x$ and $\varphi_2(x)=\exists y (y\cdot y=x)$ in the language of ordered rings $\{0,1,\leq,+,\cdot\}$. Syntactically, they are very different: the first one is quantifier-free, while the second one is not. The first one uses the symbol $\leq$, while the other one doesn't. However, if we consider them in the field of real numbers (or any real closed field), then they are equivalent: any real number satisfies $\varphi_1$ (in the real numbers) if and only if it satisfies $\varphi_2$.
More generally, if $T$ is any first-order theory in a language $L$, while $\varphi_1(x)$ and $\varphi_2(x)$ are arbitrary $L$-formulas with the same free variables, then we say that $\varphi_1$ is equivalent to $\varphi_2$ under $T$ if $T\vdash \varphi_1\leftrightarrow \varphi_2$, that is, $T$ proves that $\varphi_1$ and $\varphi_2$ are equivalent.
Essentially by completeness, this is exactly equivalent to saying that in any model of $T$, the interpretations of $\varphi_1$ and $\varphi_2$ (i.e. the sets of elements satisfying them) are exactly the same. It follows that equivalence under $T$ is an equivalence relation on the set of all $L$-formulas in variable $x$ (or any other prescribed set of free variables), and that (in models of $T$), all the semantic content of a formula is expressed by its equivalence class. For example, if we are in a real closed field, then $\varphi_1(x)$ and $\varphi_2(x)$ express the same thing, so if we only care about the meaning of the formulas, there is no need to distinguish between them. This idea gives rise to the notion of the Lindenbaum-Tarski algebra, i.e. the Boolean algebra of all equivalence classes of formulas with prescribed free variables.
In fact, we can (and do) go one step further. One can show that if we have any model $M$ of a complete theory $T$, then we can find a larger $L$-structure $N$ (a so-called elementary extension, which means, intuitively, that $M\subseteq N$ and anything we can say about an element of $M$ is true in $N$ if and only if it is true in $M$) which has the property that two elements of $N$ satisfy the same $L$-formulas if and only if there is an automorphism of $N$ which takes one of them to the other (we say that $N$ is strongly homogeneous). This reduces the problem of checking which formulas a given element satisfies to the problem of checking which orbit of $\operatorname{Aut}(N)$ it belongs to.
Moreover, if we pick $N$ to be sufficiently rich (whence it is called a monster model), virtually all interesting properties of $T$ are captured by $N$, its automorphism group, and the various Lindenbaum-Tarski algebras, or rather, their interpretations in $N$ (which we call definable sets).
$\dagger$ sometimes we do care about syntax because it tells us something about semantics, or makes it easier to understand them. For example, the theory of the real field in the language of ordered rings has quantifier elimination. This is a syntactic property, but it implies that the theory of real closed fields (in any equivalent language, for example the language of rings) is model complete, which is an important semantic property.
