Translation of sick leave

What is the logic that says that “Estou de baixa médica” means “I am on sick leave”?

Indeed, “sick leave” is the translation of “baixa médica”, a leave for medical reasons. Similarly, when a doctor discharges a patient from a hospital or confirms that they are healed or otherwise good enough to return to work/regular life (e.g. end Covid isolation), that’s called an “alta médica”.

There’s an expression, “dar baixa”, which can be used in a number of contexts, such as “dar baixa de um produto”, which refers to removing a product from the available stock or “dar baixa de uma matrícula”, which means cancelling a vehicle registration (license plate). So, in Portuguese, it’s indeed logical for us to think of “baixa” in reference to something or someone that is temporarily or permanently unavailable.