Turning on planetary automation
When you turn on automation for a planet, the planet will automatically develop new buildings and districts as needed based on population. (note: you cannot turn on automation for your capital/home planet)
To turn on automation for a planet, go to the Planetary Summary panel for that planet and click the "Automation: On" button.
You can roughly direct what types of development will be built for this planet by clicking the "Colony Designation" button to the left of this and selecting a designation.
How stockpiles work
When an automated planet needs to build something, it will draw the resources from the sector local stockpile, if there are any available, otherwise it will draw resources from the shared stockpile. If there are no resources available in either stockpile, the planet will not build anything.
Stockpiles contain a generic resource that can be used by the planet automation as minerals when developing buildings and districts. (I'm still not certain how strategic resources play into this, but I believe automation bypasses stockpiles for them and draws them directly from your strategic resource reserves.) When adding resources to a stockpile, you can choose to convert energy or minerals to stockpile reserves. The planet automation will then use the generic "stockpile" units as minerals when building.
It's also important to note that YOU CANNOT WITHDAW RESOURCES FROM A STOCKPILE ONCE THEY HAVE BEEN ADDED. Once you have converted resources to a stockpile, they can only be used by automation for development. So make sure you are not adding resources to a stockpile that you may need manually use later.
Contributing to stockpiles
To manage your sector stockpiles, you need to open the Planets and Sectors pane. (it's the fourth option in the navigation bar) The sections for managing stockpiles will all be on the right-hand side of this panel.
Local stockpiles are resources that are available ONLY for that sector. Any planets that have automated building turned on in that sector will draw from the sector's own local stockpile first. You can add resources to a local stockpile by clicking the "Transfer Resources" button. This will pop up a "Transfer Resources" window that will allow you to add resources to the local stockpile in increments of 100.
The shared stockpile is available to planets with automation turned on in ANY SECTOR, and as mentioned above, will only be drawn from if the local stockpile for the planet's sector is already exhausted. You can add directly to the shared stockpile by clicking the "+" icon under the "Shared Stockpile" section in the upper-right area. This will pop up a "Transfer Resources" window that will allow you to add resources to the shared stockpile in increments of 100.
You can also choose to contribute to the shared stockpile on a monthly basis. To do this, use the "+" and "-" buttons in the "Monthly" section in the upper-right. This will allow you to convert energy and minerals to the shared stockpile on an ongoing basis instead of needing to manually add directly to the shared stockpile.
One last thing to note is that if planet automation is short on local resources (but they are not fully depleted), it will pull the additional resources needed to develop buildings & districts from the shared stockpile. For instance, if it needed to build a City District, but there are only 300 units available in the local stockpile for that sector, it will use those 300 units and draw an additional 200 from the shared stockpile.
Please feel free to give me clarification on any of this and I'll update the answer!