Calculating Pull-Back of a $1$-form.

A $1$-form belongs to the dual space of the tangent space (at a point $p\in U$, say), that is $(T_{p}U)^{\ast}$. Hence its elements (the $1$-forms) are linear maps $\omega_{p}: T_{p}U \rightarrow \mathbf{R}$ which vary smoothly to get a family of $1$-forms $\omega:TU \rightarrow \mathbf{R}$ (i.e. I just drop the $p$ subscript). To be explicit, for some tangent vector $v\in T_{p}U$, we have that $\omega_{p}(v) \in \mathbf{R}$, or again as one varies the point to get a vector field $V\in TU$, $\omega(V)\in \mathbf{R}$.

Now given a smooth map $f:U\rightarrow V$, its differential at a point $p\in U$ is a linear map $d_{p}f:T_{p}U\rightarrow T_{f(p)}V$ which when one varies the point $p$, is usually written as $f_{\ast}$ (called the pushforward of $f$). This in turn induces a dual map $f^{\ast}:(TV)^{\ast} \rightarrow (TU)^{\ast}$ defined as follows: for a $1$-form $\alpha\in (T_{f(p)}V)^{\ast}$ we get a new $1$-form $f^{\ast}\alpha \in (T_{p}U)^{\ast}$ by precomposition, i.e. let $v\in T_{p}U$ then $$ f^{\ast}\alpha(v)|_{p} = (\alpha \circ f_{\ast})(v)|_{p} = (\alpha \circ d_{p}f)(v)|_{p} = \alpha(d_{p}f(v))|_{f(p)} $$ where $(\alpha\circ d_{p}f)$ is ''at $p$'' since $v\in T_{p}U$, yet $\alpha$ is ''at $f(p)$'' because now $d_{p}f(v)$ belongs to $T_{f(p)}V$. This construction can then be extended to $k$-forms.

To get to answering your question, $df_{(r,\theta)}(v)$ for $v\in T_{(r,\theta)}U$ hasn't appeared yet since the $k$-form is not being evaluated on any vectors (otherwise you would just get a real number). The $\omega(f(p))$ part of $\omega(f(p))(d_{p}f(v_{1}),\ldots d_{p}f(v_{k}))$ just means that your $k$-form is at the point $f(p)$, and that no vectors are being eaten up by it. In my notation above it would be $\omega|_{f(p)}$ so distinguish between being an argument of the differential form and referring to the point it is associated to. Apologies if this seems like a rather long-winded answer - a lot of the introductory theory of differential forms is unwinding definitions and remembering what spaces things live in.