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When we derive Snell's law using Huygen's principle, after refraction, we draw a perpendicular line from the point of intersection of the second light ray with the refracting surface to the first, writing the distance intercepted on the first light ray as $ v_2t $.

I'm having trouble understanding the reason behind the intercepted distance being $ v_2t $. I know it has got something to do with the the two points on the respective light rays lying on a wavefront, but I cannot fully grasp the reason.

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Here is a diagram that Huygens produced (page 35) to illustrate what happens when light is refracted.

enter image description here

CHHHA represents a wavefront part of which is about to enter the block travelling at a speed $v_1$.

LKO represent wavefronts part of which have entered the block where the speed of the wave is $v_2$ which is less than $v_1$

BN is a wavefront which is entirely in the block but would have been wavefront BMMMO if the block had not been there.
In the time that point N on the wavefront in the block travelled at distance AN it would have travelled a distance AO (which is equal to CB) if the block had not been there.

You have two triangles ABC and AOB with a common side AB.

$CB = v_1t$ and $AN = v_2t$ so now Snell's law can be derived in terms of the sines of the appropriate angles and the velocities in the two different mediums.

This is a nice simulation which might help?

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