It doesn't. Refusing to police the timeline does.
To answer your question directly, killing He Who Remains changes nothing about the timeline as it exists at that moment. The problem is that the TVA polices and prunes the timeline so that it doesn't branch and create new, alternate timelines.
The show makes it clear that the universe is chock full of possible branching points and that new branching points pop into existence all the time. We know that this can happen due to people time traveling as illustrated by Loki and Sylvie throughout the show but it also seems plausible for variants to be created entirely on their own as a natural occurrence.
Either way, once He Who Remains is out of the picture, no one is left to dispatch the TVA to prevent new nexus events. The timeline branches rapidly from there and the TVA is sent into a scramble.
Thus, the multiverse is born!
This answer does a good job explaining that there have been some mixed signals from Marvel as to how the changes to the timeline are affecting the TVA but the fact that there are changes to the TVA at all actually makes a lot of sense when you think about it.
The TVA is composed entirely of variants, as is Kang himself. If we follow Kang through this process, every time the timeline branches, a new universe comes into existence with its own potential Kang variant. That variant follows the same path that He Who Remains outlined in his conversation with Loki and Sylvie:
- He discovers the multiverse.
- He finds a way to communicate with his own variants.
- They share technology.
- They wage war.
- The winner establishes the TVA to secure peace.
It's entirely expected that a new Kang with a new pool of variants to recruit from is going to create a new TVA.