How does the battery create an electric field inside the conducting wire connected to its terminal?

if there is no wire you have an electric field between the poles of the batterie. If you connect the poles with a wire, a current will flow, driven by the electric field and the electric field is only inside the wire, outside you will have a magnetic field around the wire.

In books, it is not explained exactly how does the battery creates electric field inside the wire.

A @trula explained, the battery creates an electric field between its terminals. But with nothing connected between the terminals (infinite impedance) this field cannot produce a current. A wire with a resistance $R$ connected between the terminals allows a current $I$ to flow where $I=V/R$ where $V$ is the battery voltage across the terminals when delivering current $I$.

Also, is that the electric field inside the wire only or is it present outside the wire as well?

This question has already been asked here: Does a current carrying wire produce electric field outside?

The reference cited in the accepted answer says there is an electric field outside the wire.

Hope this helps.