I turned brown and crispy during that heat wave!

I would leave them alone and just water them thoroughly for the rest of the growing season and see what happens.

I'm not an astilbe expert, but here's my take on this ... and I assume others may not agree.

If the foliage is crisp and brown - really brown, not just brown leaf tips - it's not photosynthesising and can be removed, just to make the plants look tidier. If they're watered, the should produce new foliage. I'd leave any green stems, and just remove the parts that are truly dead. Brown crispy leaves don't generally turn green again, so if the plants are going to grow it will be by starting fresh.

they will be fine, it's typical for astilbe to get crispy and dry up in drought situations. Give them a deep watering and they will bounce back when situations improve for them.