Could you help me to understand how DNA replication works?

In order to answer your question let's separate replication process into steps.

Step 1: Unwinding of double-stranded DNA

The first step of DNA replication involves binding of helicase enzyme that unwinds and separates double-stranded DNA molecule into 2 single-stranded DNA pieces by breaking glue (hydrogen bonds) between them and creating a replication fork of two strands running in antiparallel directions.

Step 2: Making sure that DNA stays single-stranded

The single strands of DNA are sticky and want to come back together, thus in order to ensure that this doesn’t happen, single-strand DNA binding proteins attach to them that not only prevents them from binding together, but also protects them from digestion by nucleases.

Step 3: The initiation of DNA replication

In order to start the synthesis of new DNA strand, the short template strand is required on both strands of the single-stranded DNA. This is done by DNA primase that generated a short RNA primer on each of the template strands that provides an initiation point for DNA polymerase III that can only extend the nucleotide chain, but not start one from scratch.

Step 4: Synthesis of new DNA strand

DNA polymerase III enzyme attaches to the 3’-end of the primer produced by the DNA primase and covalently joins the free nucleotides together in a 5’ → 3’ direction (free nucleotides align opposite their complementary base partners (A = T ; G = C)). As DNA strands are antiparallel, DNA pol III moves in opposite directions on the two strands:

On the leading strand, DNA pol III is moving towards the replication fork and can synthesise continuouslyOn the lagging strand, DNA pol III is moving away from the replication fork and synthesises in pieces (Okazaki fragments)

Step 5: Connecting Okazaki fragments together on lagging strand

As you can see above lagging strand is being synthesised in series of short fragments containing multiple RNA primers along its length. These RNA primers are removed and replaced with DNA nucleotides by DNA polymerase I.

Step 6: Ligation of lagging strand

The final step involves DNA ligase enzyme that joins Okazaki fragments together to form a continuous strand by covalently joining the sugar-phosphate backbones together with a phosphodiester bond.