From what I understand, a $p$-form $\omega$ on a manifold $M$ is a $C^{\infty}(M)$ multilinear map $$\omega:\mathfrak{X}(M)\times \ldots \times \mathfrak{X}(M) \rightarrow C^{\infty}(M).$$
I'm my notes it says that we could also view such a form as a section of the bundle $\Lambda^pT^*M$. A section of $\Lambda^pT^*M$ is a map $s:M\rightarrow \Lambda^pT^*M$ such that $\pi \circ s =id_M$. But I do not see how to reconicle these two ideas. My understanding of the space $\Lambda^pT^*M$ is shaky.
I think the word "alternating" should be somewhere in the description you have. Anyways, I think proving the full correspondence might require slightly more serious work, but let me give you one direction of the correspondence: suppose we have a global section $s:M\to\Lambda^p T^*M$, and let me show you how to obtain such a $C^{\infty}(M)$-multilinear map of the form you describe.
By definition, for each $x\in M$ we have an element $s_x=s(x)\in\Lambda^p(T_x^*M)$. Now it's convenient to recall how exterior powers work: they are defined in such a way that for an arbitrary vector space $V$, an element of $\Lambda^p(V^*)$ is equivalent to giving an alternating $\Bbb R$-multilinear map $V\times\cdots\times V\to\Bbb R$ (where there are $p$ copies of $V$ on the left), so for our special case of $V=T_xM$, we see that every $x\in M$ is really giving us an alternating $\Bbb R$-multilinear map $T_xM\times\cdots\times T_xM\to\Bbb R$, which we will also denote by $s_x$.
Now let's recall quickly the $C^{\infty}(M)$-module structure on $\mathfrak X(M)$ are naturally a $C^\infty(M)$-module: if $V\in\mathfrak X(M)$ and $f\in C^\infty(M)$, i.e. $f$ is a smooth map $M\to\Bbb R$, then for any $x\in M$ $f(x)\in\Bbb R$ and $V(x)\in T_xM$, so the scalar multiplication $f(x)V(x)\in T_xM$ makes sense and thus we get a section $f{\cdot}V:M\to TM$ by taking $x\mapsto f(x)V(x)$, which is smooth as long as $f$ and $V$ are.
So now with this in mind, we notice that if $s$ still denotes a global section of $\Lambda^p T^*M$, then for any $V_1,\dots,V_p\in\mathfrak X(M)$, we can then induce an element $C^{\infty}(M)$ as follows: if $x\in M$, then $V_1(x),\dots,V_p(x)\in T_xM$, then because $s_x$ determines an alternating $\Bbb R$-multilinear map $T_xM\times\cdots\times T_xM\to\Bbb R$ as we wrote above, we see that we can apply this to our elements $V_1(x),\dots,V_p(x)$ to get $$s_x(V_1(x),\dots,V_p(x))\in\Bbb R$$
Now we can think of this as giving us a function
$$s:\mathfrak X(M)\times\cdots\mathfrak X(M)\to C^\infty(M),\quad (V_1,\dots,V_p)\mapsto s(V_1,\dots,V_p)$$
where $s(V_1,\dots,V_p)$ is the function $x\mapsto s_x(V_1(x),\dots,V_p(x))\in\Bbb R$. If you use our definition of the $C^\infty(M)$-linear structure, it will be immediate to check that this is $C^{\infty}$-multilinear and alternating because each $s_x$ is $\Bbb R$-multilinear and alternating.