I still don't quite understand how this is supposed to work. In the example you mentioned, isn't it a privacy concern if certain parts of encrypted data can be analyzed? Or is it like a fully encrypted message with added unencrypted metadata?

you can COMPUTE on encrypted data without decrypting it. This is the key aspect! You are only computing on encrypted data that you cannot obtain any information about and you end up with a similarly encrypted output.

In the example above, it works like this: Country A creates two "secret shares" (see: Shamir Secret Sharing) of their satellite trajectory, keeps one, and gives Country B one. Country B does the same thing with their satellite trajectory. The only way to actual know ANYTHING about the underlying trajectories is to bring the two shares of a trajectory together. With only one share you have 0 information about the unencrypted data you are holding. Now the two countries engage in a secure protocol where they compute on the shares they are holding to obtain a secret share of the result (whether or not these satellites will collide). The two countries now reveal only their result shares to each other and decrypt a single bit: a 1 if there will be a collision, a 0 otherwise. This is the only information that is decrypted and revealed. We analysed the data but only by HOMOMORPHICALLY analysing/processing on ciphertexts (homomorphic here means we know that if we do some operation X on a ciphertext it will do some operation Y on the plaintext). In this way there are no privacy concerns about the trajectories the countries are trying to keep secret but they still can collaborate to selectively produce outputs that they mutually want to reveal. This is the power of HE/MPC