1) The left hand figure is a traditional pendulum, and the period is dependent on the length of the pendulum. The right hand figure uses the quanity 'L' to help define the value of the spring constant  the assumption being that under weight mg the beam deflects distance L. The period of a typical springmass systems is 2 pi sqrt(m/k), but in this case you have a rotating beam and taking into account the rotational inertia and torques being applied the period turns out to be 2 pi sqrt(m/4k). Replace k by 4mg/L and you get the equation provided in the figure.
2) Because it works top make both fomulas the same. If the weight mg was placed at the same point as the mass teh equation woulw end up being T= 2 pi sqrt(L/(2g)).
