us
咱 centers on the first-person plural: us or we. It is a rare character whose standalone meaning is simply this collective self-reference.
咱 combines 口 (mouth) with 自 (self), suggesting a spoken self-reference that includes others. The exact historical development is uncertain.
The mouth 口 speaks for the self 自 and others together: 咱 means us.
For サツ, imagine a group of us sitting together on a mat: sat -> サツ, and we are all seated as one.