Why is the water dripping from my patio furniture removing the sand from the joints between bricks?
It's going to take a few more re-applications of sand and more rain, until it will do it less. That's exactly what I'd expect it to look like after the first rain. 'Sand' it again.

Using a push broom and a garden hose, work the sand into the cracks. I'm guessing you skipped hitting it with water the first time, or the sand is too course to set properly. As those are some really tight tolerances on the gaps (nice work), I can see having a hard time getting it in there.

Work the problem areas with a stiff bristle brush while still letting water run over it, eventually they will fill.

If my assumptions about it not being filled all the way in the first place are wrong and it is getting washed away, once it becomes (grout) a mixture of dirt and what-have-you, it should do that much less.