Most significant bits (plural) are the bits closest to, and including, the MSB. Similarly for LSBs. A simple 13 bit binary counter will overflow exactly 189 times per second at 1.548288MHz (every time all 13 of those trailing zeros appear simultaneously) . That sounds like an odd number but it factorises into usable numbers for gearing.
It may be easier to visualise with lower numbers. 40Hz (not a power of 2) will overflow a 3 bit counter exactly five times a second - every time the counter passes 1112. Converted to base-10 that happens at counts of 8, 16, 24, 32 and 40.
This could all be absolute nonsense and it's achieved in a completely different manner, but that way seems fairly straightforward.