I just checked my favorite reference, and it goes through six pages of how without giving any why. I expect that there's some multi-page dive into linear algebra that shows it explicitly, and boy do I wish I had access to that dive.
But, here's two intuitive arguments for why it works:
First, the Kalman filter is optimal for the information it has been given. When you do a prediction step, the result is an optimal guess, based on the prior state. When you do a correction step, the result is optimal given linear combinations of states you've measured.
So, when you do a sequential correction, the correction on the first measurement is optimal for that measurement. The correction on the second measurement must be optimal for that measurement and the first one. And so on and so on, until you're done and you have a correction that's optimal for all of the measurements you have. That's the same as you'd get doing the correction all in one step, so it has to match.
This is the "just trust the assertions of the math" argument.
Second, and this is definitely a loose intuitive argument, any single-point measurement is going to narrow down the possible values of the states in exactly one direction in the states space. I.e., a measurement $y = \begin{bmatrix}1 & -1 \end{bmatrix} \mathbf x$ tells you a lot about the linear combination $x_1 - x_2$, but it tells you nothing about $x_1 + x_2$.
So as you sequence through your measurements, you'll be correcting the states first in one direction, then another, then another, until finally you're done. Note that this is sorta-kinda what you're already doing in a Kalman filter that has a lower dimension measurement than the number of states: if it's going to work at all, each iteration of the prediction step rotates the $\mathbf x$ vector (and, by extension, the $\mathbf P$ matrix), so even as each correction step only ever corrects in one direction, the result is still an overall improvement in knowledge over the whole state space.
