Suppose I have a $k$-dimensional manifold embedded in $\mathbb{R}^n$. Munkres defines a $k$-form on $M$ as a a function $\omega$ that assigns an alternating tensor at each point $p \in M$ that acts on $k$-tuples of tangent vectors in $T_pM$. He says that, for simplicity, we will just work with $k$-forms that are defined on open sets (of $\mathbb{R}^n$) which contain $M$, as we can restrict this $k$-form to $M$.
So if we have a $k$-form $\omega$ defined on an open set of $\mathbb{R}^n$, then we can write $\omega = \sum f_I dx^I$ where $I$ represents an ascending $k$-tuple of integers from the set $\{1, \dots, n\}$ and the $f_I$ are smooth functions. However, I would like to think (and I may be wrong here) that if we really want a $k$-form $\omega$ defined on just $M$, then because $\dim(M) = k$, we ought to be able to write $\omega = f dz^1 \wedge \cdots \wedge dz^k$, where the $z^i$ are standard coordinates on the tangent space at each point. Being able to write a form in that manner seems much more desirable to me. I was told however, that in general, the tangent spaces of the manifold will be in different directions, so I cannot just declare $z^i$ for $1 \leq i \leq k$ to be standard coordinates on the tangent space, for the $z^i$ will, in general, be different for each tangent space. This makes sense to me.
However, this also bothers me because why should a $k$-form be dependent on the space that the manifold is embedded in? By writing $\omega = \sum f_I dx^I$ we are implicitly making reference to the ambient space. There ought to be an intrinsic way to define a $k$-form on a $k$-manifold. Also, since $\omega$ acts on $k$-tuples of tangent vectors, and the tangent space can be defined via derivations without reference to an ambient space, then $\omega$ ought to take on the same values, regardless of what space $M$ is embedded in. At least these properties seem reasonable to me.
On the Wikipedia page, there is an intrinsic definition: https://en.wikipedia.org/wiki/Differential_form#Intrinsic_definitions
But I don't know enough smooth manifold theory yet to understand that definition. My only workings with smooth manifold theory so far have come from Munkres and the first three chapters of Lee's "Introduction to smooth manifolds" (which covers smooth manifolds, smooth maps between smooth manifolds, tangent spacces). So I was hoping that someone would be able to explain it in an intelligible way to me. Also, please correct anything that I may have gotten wrong, and definitely correct me if my list of "reasonable properties" is wrong as well. Thanks.
Your intuition is correct, you can do that without embedding a manifold $M$ into Euclidean (or other) ambient spaces. The basic tool you need for this is (obviously) a definition of (smooth) manifolds which does not assume an embedding into such a space. Local charts/coordinate systems, the domains of which cover $M$, are what is needed for this, you may have encountered them already when reading the books you mentioned.
The basic idea is to locally represent the manifold as diffeormorphic image of a Euclidean space of the same dimension of $M$. If you do that you will have regions of overlapping coordinate patches, for these you need compatibility conditions (the change of coordinates has to be as smooth as you intend the manifold to be).
The idea is to use this same construction to define the tangent space on $M$, by starting out with a model defined by coordinate patches and by ensuring it's well defined by requiring coordinate changes to transform these spaces in a consistent manner. It turns out that this is, in principle, possible, leading to something called the tangent bundle of a manifold, which you can think of as assigning a tangent space (vector space having the same dimension as $M$) to each point of $M$ in a consistent way. (In case of an embedding you may think of this as a plane attached to $M$ in each point in a smooth manner). The construction as such is not difficult but a bit involved from a technical point of view.
With this construction you can then carry out all kinds of algebraic constructions with the tangent space, e.g. look at the dual space or tensor products of the tangent and dual. In particular you can define $k$- forms on this construction (this is what the Wikipedia article is about).
Again, this is technically involved cause you need to verify that this is well defined, independent of, e.g. the choice of coordniate system, but it's a very powerful and successful approach.
Another consistency question then arises if you are looking, again, at embedded submanifolds of Euclidean spaces. You then need to verify the various definitions you have for the same concepts do agree then.
Almost any modern textbook on differential Geometry should provide the details of this.