How does a cell use endocytosis to form a vesicle

Endocytosis is an active transport system. Vesicles have a membrane similar to the phospholipid membrane of a cell. When a solid material enters a cell, the plasma membrane forms a U-shaped membrane and surrounds the solid particle. This is done because of Pseudopodium. Eventually, the pseudopodium completely surrounds the solid particle and forms a phagosome, which is a vesicle. This is similar for fluids, too. However, there is another type of endocytosis known as Receptor-mediated endocytosis; this is a bit more complicated and for middle school, it is not required to know this. But in case if you are curious, here is what happens: certain receptors surround the cell. The receptors cling to particles and transfer the particles inside with the help of coat protein. 

In brief, there is a process called Pinocytosis or endocytosis, which is how vesicles are formed. Solid and fluid particles are enclosed by the plasma membrane and are brought into the cell. Eventually, a small spherical shape is formed, similar to a bubble. Eventually, that piece of plasma membrane splits off from the cellular membrane and becomes the vesicular membrane. It also has phospholipids, where the lipids are hydrophobic and the phosphorus is hydrophilic. 

I hope this helped!