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The world lines of both ends of the metre stick are consequently vertical. Now consider another primed frame moving at velocity v relative to the stationary frame, calibrated such that both frames measure a time of 0 when the origins coincide.

The lines corresponding to x' = 0 and t' = 0 are drawn on the spacetime diagram below (from the Theoretical Minimum lecture series)

He says that the length of the metre stick in the moving frame of reference is represented by the distance from the origin to the intersection of the lines x = 1 and t' = 0 (between the two dots). I can't understand why this is the case.

It makes some sense to imagine that the length of the metre stick in the primed frame will be obtained from setting t' = 0 and measuring the x' coordinate at the end of the metre stick, but what has intersection with the x = 1 line got to do with this - that is, what is stopping us from labelling the position of the end of the metre stick anywhere else along the t' = 0 line?