Question

what is the function of vesicles in the synthesis of proteins and the release of those proteins outside the cell

Answer 1

Proteins that are meant to be secreted out of the cell or to be embedded into the plasma membrane are synthesized into the endoplasmic reticulum. From there, they undergo various modifications in the Golgi apparatus where they are eventually "packaged" into vesicles and undergo exocytosis. The vesicles contain a membrane that is essentially the same as the plasma membrane. This allows for it to merge with the plasma membrane from the inside of the cell and release it's contents into the extracellular space .
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