Neat!

Someone in the comment section was also kind enough to provide a fairly rigorous breakdown of how it actually works:

Let √c = a + b, where a represents the whole number component of the answer, a ∈ Z, and b represents the decimal component, 0 < b < 1
c = (a + b)²
c = a² + 2ab + b²

Now solve for b:
2ab = c - a² - b²
b = (c - a² - b²)/2a

b² (the decimal component squared) will be very small, so we can ignore it therefore, b ≈ (c - a²)/2a
in other words, b ≈ the original number, c, minus the nearest perfect square below it (the whole number component squared), all over 2a: which is the formula given in the video for working out the non-whole number part of the answer e.g.

√27 = 5 +b 27 = (5 + b)² 27 = 25 + 10b + b²

10b = 27 - 25 - b² b = (27 - 25 - b²)/10

b ≈ (27 - 25)/10 b ≈ 0.2 √27 ≈ 5 + 0.2 = 5.2

As further proof, the approximate answer will always be out from the actual answer by a margin of b²/2a: 5.2 - 5.196... = (0.196...)²/10