A covector basis \(\left\{ \pmb{\theta}^i \right\}\) is defined to act on the corresponding vector basis \(\left\{ \pmb{e}_j \right\}\) in the manner,
\[\pmb{\theta}^i \left( \pmb{e}_j \right) = \delta^i_{\phantom{i} j}\]Where \(\delta^i_{\phantom{i} j}\) represents the Kronecker delta. But where does the above definition even come from? Well, turns out it’s not so mysterious after all.
Say a covector \(\pmb{\phi}\) acts on a vector \(\pmb{x}\). In the component form,
\[\pmb{\phi} \left( \pmb{x} \right) = \left( \phi_i \: \pmb{\theta}^i \right) \left( x^j \: \pmb{e}_j \right)\]By multilinearity,
\[\pmb{\phi} \left( \pmb{x} \right) = \phi_i \: \pmb{\theta}^i \left( \pmb{e}_j \right) \: x^j\]We want the result of \(\pmb{\phi} \left( \pmb{x} \right)\) to be invariant. The simplest way to do so is to define it to be a scalar of the form \(\phi_j \: x^j\). Therefore,
\[\begin{align} \phi_i \: \pmb{\theta}^i \left( \pmb{e}_j \right) \: x^j & = \phi_j \: x^j \\ \phi_i \: \pmb{\theta}^i \left( \pmb{e}_j \right) \: x^j & = \phi_i \: \delta^i_{\phantom{i} j} \: x^j \end{align}\]By linear independence of the components involved, we can cancel out like terms on both sides of the above equation, and are left with the required result,
\[\pmb{\theta}^i \left( \pmb{e}_j \right) = \delta^i_{\phantom{i} j}\]