The exact sequence is defined as follows:
$$0\rightarrow \mathbb{Z}[t]\xrightarrow{\binom{1-t}{p}}\mathbb{Z}[t]\oplus\mathbb{Z}[t]\xrightarrow{(p, t-1)}\mathbb{Z}[t]\xrightarrow{\eta}\mathbb{F}_p\rightarrow 0$$
These are all $\mathbb{Z}[t]$-modules. The morphism is defined to be the usual multiplication and $\eta(t)=1$. For example, if $f(t)\in \mathbb{Z}[t]$, then $\binom{1-t}{p}(f(t))=\binom{(1-t)f(t)}{pf(t)}$. If $\binom{f(t)}{g(t)}\in \mathbb{Z}[t]\oplus\mathbb{Z}[t]$, then $(p,t-1)\binom{f(t)}{g(t)}=pf(t)+(t-1)g(t)$.
The goal is to compute $\text{Ext}^k(\mathbb{F}_p, \mathbb{Z}[t])$. I can't work out the computation details. Could someone give a hint?