The notion of differential forms depends on several structures:
- the wedge product
- the dual space
- the tangent bundle
- sections
So its not suprising that your lecturer wasn't as easily able to motivate them as vectors! Lets take these step by step:
1. The wedge product
In a 3d vector space we have the additional structure of an inner product and the cross product. These have geometric interpretations. However, when we generalise to a vector space of any dimension it's easy to see that the inner product generalises in an obvious way. Not so the cross product. In fact, this is only available in 3d.
Recall, that the scalar triple product $u.(v \times w)$ gives the volume of the parallelopid formed by the sides $u,v,w$. It is this property that generalises.
Given a parallelepid in a n-dimensional vector space $V$ (this is the generalisation of a parallelogram in the plane) whose sides are $v_1,..., v_n$. Then the wedge product $v_1\wedge ... v_n$ gives us the signed volume. It turns out that this is a vector, but they don't lie in the same vector space as $v$. We call them $k$-vectors and say that $v_1\wedge ... v_k$ lies in $\wedge^k V$.
2. The dual space
The dual space of a vector space $V$ is usually written $V^*$. It consists of all linear functions to the real line, $f:V\rightarrow R$. What does this mean? Each function is linear, so we can think of it as a kind of measurement or metric on the vector space. It tells us how to measure a vector. Thus $V^*$ is the space of all the ways we can measure vectors in $V$.
3. The Tangent Space
Given a manifold $M$, we can construct its tangent bundle $TM$. The easiest example to visualise is when the manifold is a curve or surface. Lets take the curve first: at every point of the curve $C$ we can draw the tangent line to it, this line extends to infinity and is a 1d vector space. We bundle them up all together into the bundle $TC$, and the tangent line at the point $p$ on the curve is $T_pC$. Similarly for a surface $S$, at each point $p$ of the surface we can draw the tangent plane to it, we write this as $T_pS$ and we bundle them all together into the bundle $TS$.
Now any bundle $E$ over a manifold $M$, has a projection map $\pi:E\rightarrow M$ and this is how they are usually referred to. It tells us where the 'fibres' are attached to. If we take the first example, $TC$, the tangent bundle of the curve; let $v$ be a vector in one of the tangent spaces, say $T_pC$ - this means that $v$ is in the tangent line (rather vector space) - that is defined (or attached) to the point $p$ of the curve. The projection map $\pi$ simply maps $v$ to the point $p$. So we can see that the image of the entire space $T_pC$ is just the point $p$.
4. Sections
Given a bundle $\pi:E\rightarrow M$ then we can take its space of sections $CE$. This is the space of all maps $s:M\rightarrow E$ such that $pi\circ s =Id_M $. For example, suppose $E$ was a bundle of vector spaces over the manifold $M$, then a section is a choice of a vector in each fibre. It is a vector field.
Construction of differential forms
Finally we put all these structures together: We construct the bundles $\wedge^k T^*M$. That is we take the manifold $M$, we construct the tangent space $TM$ over it, and then take it's dual space $T^*M$ and the finally we take the $k^{th}$ wedge $\wedge^k T^*M$. The sections of this bundle is $C(\wedge^k T^*M)$ and this is the space of all $k$-differential forms and is usually written (at least by mathematicians and sometimes others) as $\Omega^kM$.
Uses
It turns out that we have a map $d^k:\Omega^kM \rightarrow \Omega^{k+1}M$ called the exterior derivative (another name for the wedge product is the exterior product) and this generalises the $grad$ operator in vector analysis. That is $d^0=grad$. The other vector analysis operators - $div$ & $curl$ - are variants of this.
It also turns out that when we integrate a form $\omega$ over a manifold $M$ we get a generalisation of Stokes theorem: $\int_M d\omega=\int_{dM} \omega$, where the symbol $dM$ is the boundary of the manifold.
conclusion
Thus we see that differential forms allow us to generalise the vector analysis that we're already familiar with in 3d Euclidean space to the context of manifolds of any dimensions. This is important given the importance of vector analysis in physics. But they have many other uses, for example de Rham cohomology. They also bring in many other notions that are important, for example vector, fibre and principal bundles.
There is a formulation of General Relativity that uses a connection on the frame bundle of the tangent bundle and this a principal bundle with structure group the Lorentz group. This connects to the way the other forces are described, for example electromagnetism, the electroweak and the strong force are described as principal bundles with structure group $U(1),SU(2)$ and $SU(3)$ respectively in the Standard Model.
