When mRNA is synthesized in the nucleus during the process of transcription, it needs to be transported to the cytoplasm for translation into protein. The mRNA molecules exit the nucleus through specialized channels called nuclear pores. These nuclear pores are large protein complexes that span the nuclear envelope, which is the double membrane surrounding the nucleus. They serve as gateways allowing the selective passage of molecules like mRNA, while preventing free diffusion of other molecules between the nucleus and cytoplasm.

To clarify the other options:
- The endoplasmic reticulum (ER) is involved in protein synthesis and processing but is located in the cytoplasm and does not serve as a passageway for mRNA from the nucleus.
- The Golgi apparatus is responsible for modifying, sorting, and packaging proteins after their synthesis and is not involved in nuclear export of mRNA.
- Cytoplasmic vesicles mediate transport within the cytoplasm, but mRNA exits the nucleus before these vesicular pathways come into play.

Hence, the correct and primary route for mRNA to leave the nucleus is via the nuclear pores.

Important points:
- mRNA exits the nucleus through nuclear pores.
- Nuclear pores are protein complexes in the nuclear envelope that regulate molecular traffic.
- Endoplasmic reticulum, Golgi apparatus, and cytoplasmic vesicles do not mediate mRNA nuclear export.

Reference: Molecular Biology of the Cell, 6th Edition, Chapter 4: The Nucleus and Chromosome Organization, Page 218