The purpose of the extraction chambers is to collect "Diamond essence", which is needed to create new gems.
This got explicitly answered from show creator <PERSON> during the January 21 Reddit AMA (Ask Me Anything), when she, together with show writers and storyboarders <PERSON> and <PERSON>, answered question from fans:
Q: What is the purpose of blue and yellow diamonds "sweat" being collected in each scene in the episode familiar?
A: To create more Gems. This is an important part of the substance injected into kindergartens - please see "How Gems are Made" -RS
(source: reddit post)
The chambers were introduced in the episode "Familiar" (S05E26), aired 4 weeks before the AMA, which already strongly hinted at this important function.
(source: wikia)
The pool chamber is probably the one and only Era 1 extraction chamber. It bears the older insignia of the Diamond Authority and originally all four Diamonds bathed in it, but during Era 2 only Blue used it:
[Blue Diamond]: We had such fun together -- You and I and Yellow and White.
...
[BD]: We stopped coming here together after you left. 167
That was the start of Era 2. No one wanted to be here -- not with your empty spot. It was so quiet.
(source: S05E26 Familiar, wikia transcript)
In the wide shots of the location many pipes can be seen, possibly taking the precious fluid wherever it is stored or used.
Yellow Diamond's sauna is more recent, dating back to Era 2: it shows the newer Diamond Authority insignia and is probably used only by Yellow.
Toward the end of the episode <PERSON> himself, after discovering <PERSON> secret laboratory, creates a new pebble with a droplet of his sweat.
With this knowledge, <PERSON>'s fountain on Earth (shown in the episode "An Indirect Kiss" S01E24) is revealed to be very similar to an extraction chamber, but aimed at repairing cracked gems instead of creating new ones.
! The season finale "Change Your Mind" (S05E29) shows that <PERSON> and the Diamonds, bathing together in the fountain, are finally able to cure the corrupted gems, a feat that <PERSON> alone could not manage, and neither <PERSON> with Yellow and Blue. 167