Should I ask for permission to write an expository post about someone else's research?
I don't think you strictly need permission, but I would recommend asking the speaker anyway, out of simple courtesy.  People are aware when they give a talk that they are making the information public, and that the audience members may share what they have learned, but putting it on the Internet may speed up that process more than the speaker is expecting.  Posting about the talk without the author's consent, or at least notifying them, isn't unethical, but it is sort of "bad manners".

If the speaker asks you to wait, I would suggest you honor that request: partly out of  general professional courtesy, and partly because it isn't good politics as a PhD student to get on the bad side of senior people, especially if you work in this area.  You may be applying for a postdoc with this person someday.  I would guess that at most, they might say something like "We plan to post a preprint in a few weeks; could you hold off until then?"

As a possible side benefit, when you contact the speaker to ask if it's okay with them, they might also volunteer some other information for you to use in your  writeup: copies of their slides, further details, a preprint, etc.

If they say it's okay, then as a further courtesy, you may want to send them a copy of your writeup before you post it, to see if they have any comments or notice any errors.  Or at least send them a link when you post it, and be ready to respond to any feedback they may have.