Manifolds and differential forms are foundational concepts of differential topology, and connections are a foundational concept of differential geometry. They are mathematical building blocks used in modern physics, essentially enabling the transfer of multivariable calculus to arbitrary curved surfaces (without relying on an explicit embedding into Euclidean space). I think the joke is that physics students don't typically learn the details of these building blocks, rather just the relevant results, and get confused when they're emphasized.

For a tl;dr about the concepts mentioned:

• A manifold is a curve, surface, or higher-dimensional object which locally resembles Euclidean space around each point (e.g. the surface of a sphere is a 2D manifold; tiny person standing on a big sphere perceives the area around them to resemble a flat 2D plane).

• Differential forms are "things that can be integrated over a manifold of the corresponding dimension." In ordinary calculus of 1 variable, that's a suitably regular function (e.g. a continuous function), and we view such a function f(x) as a differential form by writing it as "f(x) dx."

• A connection is a way of translating local tangent vectors from one point on a manifold to another in a parallel manner, i.e. literally connecting the local geometries of different points on the manifold. The existence of a connection on a manifold enables one to reason consistently about geometric concepts on the whole manifold.