At a replication fork, the two parental DNA templates are antiparallel, but DNA polymerase can extend a new strand only $5'\rightarrow3'$. The cell solves this geometric mismatch by synthesizing the two daughter strands differently.

A primer supplies the first $3'$ end

Most replicative DNA polymerases cannot start a new strand from nothing. A primase first synthesizes a short RNA primer that provides a free $3'$ hydroxyl. DNA polymerase can then extend from that end.

Leading strand

On one template, the direction of fork opening allows DNA polymerase to follow the fork while synthesizing continuously.

This daughter strand is the leading strand.

Lagging strand

On the other template, continuous synthesis toward the fork would require forbidden $3'\rightarrow5'$ polymerization. Instead, DNA is synthesized discontinuously as short Okazaki fragments.

Each fragment begins with a new primer and is extended $5'\rightarrow3'$ away from the moving fork. As more template is exposed, another fragment begins closer to the fork.

A simplified picture is

fork movement →
leading:  continuous synthesis →
lagging:  ← fragment  ← fragment  ← fragment

The arrows indicate the $5'\rightarrow3'$ direction of each newly synthesized segment.

Fragments must be converted into one continuous strand

After synthesis, the RNA primers are removed and their positions are filled with DNA. This leaves adjacent DNA segments separated by backbone nicks. DNA ligase seals those nicks, converting the Okazaki fragments into one continuous daughter strand.

Thus both daughter strands are ultimately continuous DNA even though one was assembled in pieces.

The distinction between leading and lagging strands is not caused by different copying rules. Both use complementary base pairing and $5'\rightarrow3'$ polymerization. The difference arises solely from applying that same directional chemistry to two antiparallel templates at a moving fork.