berkeley-softfloat-3 | SoftFloat release 3 - John R | Math library

The shifts are crafted so as to have the product fitting on 63 bits available with signed integer arithmetic.

You have sigA on 24 bits, that means sigA is bounded by sigA < 2^24.

Same for sigB: sigB < 2^24.

Thus the product sigA * sigB < 2^48, or in other words, you have a guaranty that it fits in 48 bits
(it is at most (2^24-1)^2=2^48-2^25+1).
Therefore, if you shift by 15 bits (8+7), you have a guaranty that the results fits on 63 bits

Due to implied ones, you also have sigA >= 2^23 and sigB >= 2^23, thus sigA*sigB >= 2^46. IOW, it spans at least 47 bits.

So by shifting by 15 bits, you are sure that the results fits either on 63 bits, or on 62 bits. It does not matter how many time you shift each significand, 15+0 or 1+14 would work equally well. But with a well balanced 8+7, each one still fit on 32 bits which is clever.

In the next line, you see that the product is always adjusted to fit with most significant bit at position 63 (1-based):