Getting someone's "express permission" means clearly and specifically getting their permission to do something.
Sometimes it's OK to do something without getting express permission for it. For example, if you are renting an apartment you can probably hang a small picture on the wall without asking your landlord. But if you want to install new kitchen cabinets, you will need to get your landlord's express permission.
This is a formal and legal-sounding phrase.
The word "express" is also used as an adjective to mean "specific" in this phrase:
I came here for the express purpose of finding a job.