If you watched this first before “To Every You I've Loved Before”, you might get confused and find “To Me, The One Who Loved You” less appealing. Even though the production team and writer claimed that these two stories are standalone and can be watched in any order, I strongly discourage you from doing so. Watch “To Me, The One Who Loved You” first before watching “To Every You I've Loved Before”.
That said, let's go to the explanation of the ending.

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This is assuming you've watched both already, preferably “To Every You I've Loved Before” first, then “To Me, The One Who Loved You” next.

At first, the timeshift would have been useless since Koyomi chose to timeshift to himself before the divergence occurred, which was before he had to choose between his mother and father. Why? The parallel world where he picked his mother, let's call it the “Kazune-world”, will just happen the way it is. Nothing will change, even if Koyomi from the “Shiori-world” (where Koyomi picked his father) timeshifts to it.

Timeshifting doesn't change the past because (1) the shifted identity will not remember anything. Zero. Zilch. The timeshifted identity will simply merge with its past self and that's it, his future knowledge is gone. In other words, his timeshifting was useless.

However, that was not the point. Koyomi from Shiori-world realised how him and Shiori Ghost are quantum entangled. So, no matter where or when Koyomi goes, Shiori will be dragged as well, be in space or in time. The quantum physics in this multiverse is that, those that are quantum entangled must exist in the same space and at the same time (side note: in the real-world quantum entanglement, both objects can exist anywhere in space and anywhen in time).

Koyomi realised that when he dies, Shiori will be stuck as the Ghost at the Intersection, forever. So, he tried to find a way to timeshift so he can drag Shiori's ghost with him and merge her with her past version in the “Kazune-world”.

This is why the ending of “To Me, The One Who Loved You” ended with both of them going down together. Once they reach the past, Koyomi of the Shiori-world will merge with his past self and forget himself; and Shiori will also merge with her past self and forget herself.

Back in the Shiori-world, the Ghost at the Intersection will cease since Shiori is gone as well; while Koyomi will probably be in a coma for a short while until his body gives up.

Q: How come, in “To Every You I've Loved Before”, the old Koyomi from the Kazune-world saw the Shiori ghost?

It was the promise they made not to forget. It was their hope. Somehow, this part of their memory resurfaced 66 years later at that intersection, which allowed Koyomi of Shiori-world, and Shiori Ghost, to meet one final time.

Hence, when Koyomi of the Kazune-world looked at his IP watch, it was “error”. The reason is, he never parallel shifted. His personality just had a sudden interruption because the remnant Koyomi of the Shiori-world resurfaced at that moment.

Q: How can the Shiori Ghost be explained?

It could either be Koyomi of the Shiori-world's imagination; or, the Shiori who merged with the Shiori of Kazune-world, somehow was able to manifest herself at that moment.

Remember, Shiori of the Kazune-world was also at that exact same area at that exact time. Like Koyomi, who had a momentary identity interruption, Shiori more likely did as well.

Thus, when everything went back to “normal”, Shiori of the Kazune-world showed up to pick up the medicine of Koyomi of the Kazune-world. Shiori was there too. The fact that it took a few minutes for Shiori to show up, hints that she probably collapsed or fell asleep as Shiori Ghost momentarily manifested herself. Remember, the two Shiori's merged when they were kids.

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That's how the two films are connected.

As far as Koyomi of the Shiori-world is concerned, he gave Shiori a chance to live a new life, by merging her to her Kazune-world version.

Q: But wait! The timeshift was before the Koyomi divergence!

Correct. That means, Shiori Ghost and Koyomi of Shiori-world, were also in both parallel worlds. So, yes, there is a version of them that keeps on repeating the Shiori-world events.

In my opinion, Koyomi should have chosen to timeshift to the Kazune-world, right after the Koyomi divergence. Or maybe that's what he did, and the subtitle translation was inaccurate.

One last thing:
Q: When did Koyomi of Shiori-world formed his plan?
A: When Koyomi shifted again to Koyomi of Kazune-world, where Koyomi and Kazune slept together.

That was when Koyomi of Shiori-world understood what to do, and why he was willing to make the sacrifice.

Easter egg:
In “To Every You I've Loved Before”, Kazune kept on talking about the Koyomi who saved her when they were little kids. It was Kazune of Shiori-world. He did not exactly save Kazune, he was simply chasing his dog who was chasing another dog that was chasing Kazune. For Kazune, that made Koyomi kind, helpful, friendly, thoughtful, especially after she saw his affection with his dog.

They probably had a short interaction, hence Kazume also kept saying that Koyomi is this and that good qualities, but the Koyomi in high school is different.

That's basically it. ^_^