What is the cause of discontinuous synthesis of DNA on one of the parental strands of DNA? What happens to these short stretches of synthesized DNA?

The main enzyme involved in DNA replication is the DNA-dependent DNA polymerase. This enzyme catalyses the polymerisation of deoxynucleotides along the 5′ → 3′ direction, and hence, replication is continuous along the 3′ → 5′ strand (leading strand) and discontinuous along the template, i.e., the 5′ → 3′ direction (lagging strand).
The DNA replication is discontinuous in the lagging strand because as replication progresses, opening of this strand exposes the new template, hence RNA primers need to be added continuosly.

Okazaki fragments are short DNA segments on the lagging strand, formed in the 5’ – 3’ direction, starting from RNA primers. A separate RNA primer is needed for the synthesis of each Okazaki fragment. These discontinuously synthesized fragments are later joined by the enzyme DNA ligase.