Why does Leonard Susskind draw constant time slices around a black hole as lines passing through the origin at zero? In this video Inside Black Holes by Leonard Susskind,
why does he draw the constant time slice as lines passing through the origin at zero?
Something seems to be contradicting to have constant time slice meeting at the same point.

Is that effect due to being around blackhole? How precisely it is?
 A: 
Is that effect due to being around blackhole?

No, what he is showing there is the standard coordinate chart for a uniformly accelerating observer in flat spacetime. There is no black hole or spacetime curvature in that diagram. It is standard flat spacetime with accelerated coordinates instead of inertial coordinates.
Recall that the equivalence principle says that locally the effects of being in a gravitational field are the same as being in an accelerated reference frame. Susskind is drawing this Rindler diagram to explore the similarities between the features of the Rindler coordinates and the black hole.

Something seems to be contradicting to have constant time slice meeting at the same point.

Yes, you are correct. One of the mathematically essential restrictions on coordinate charts is that they must be a one-to-one mapping between events in spacetime and points in $\mathbb{R}^4$. The event at the origin has multiple coordinates in this chart, so it is not valid there. In fact, technically this chart does not include any of the events on the 45 degree lines. The border of the coordinate region is not part of the chart.
