The Observatory allows the construction of probes and colony ships at the shipyard. It also provides a list of all your active probes and the stars they are observing. A higher building level allows more probes to be constructed. You may only have one Observatory on a planet.
You can manage three probes per level of observatory. Also, your probe data is shared with the other members of your alliance. So if you form an alliance, you can create a vast network of probes that will allow you to see a large portion of the Expanse.
You will get one starter probe at your home star when you start even tho you don't have an observatory --- this is a normal probe and it can be destroyed by normal means (such as having an observatory for it and then losing it, abandoning the probe, spy actions, or detonators).
Observatories at level 15 allow making fissure sealers, to fix those otherwise planet-destroying monstrosities of not having said "what's the worse that can happen anyhow?" when playing with black hole generators.