Electrons emitted by a hot filament are accelerated by an electric...

The electric field $E$ is directed towards the left.

The force on an electron is $\vec{F} = -e\vec{E}$. Since the electron is negatively charged, the electric force acts in the direction opposite to the electric field, which is towards the right.

This force accelerates the electrons as they move from left to right. Thus, the speed of the electrons increases continuously, making the speed at $B$ higher than at $A$.

In steady state, the current $I$ is constant along the beam. With a uniform cross-sectional area $A$, the current density $J = \dfrac{I}{A}$ is also constant.

The conventional electric current flows opposite to the direction of electron motion, meaning it flows from right to left.