Why does adding a row affect other columns in Pages?

If you're using a table then, ostensibly, every row has the same number of columns, and every column has the same number of rows.

Adding an extra column will add that column to every row.

Adding an extra row will add that row to every column.

That's just how it works.

Like every great rule, though, there are exceptions. Which one's best for you depends on what you're trying to achieve.

As Viking already mentioned, you can wrap text within a cell that extends the cell height to accommodate the text.

Or you can select a range of cells and merge (Ctrl-click -> Merge Cells) them into a logical group that looks (and acts) like a single cell, such as for the siblings Sue & Oliver, and Liam and Mia's parents in this example:

Note that Column C still has 7 rows, just that cells C3 and C6 are merged with their respective preceding cells.

Merging is most often used when the value applies to multiple rows (or columns, depending on which way you merge), as in the case of Parents Mary & John.

What you can't do is have row A with 20 rows and row C with only 5 (at least without merging cells so they look like 5 cells. If you truly want different numbers of rows then consider inserting multiple distinct tables and positioning them alongside each other.

This depends on whether you want to insert another row or you want to wrap text in a specific cell.

If you want to add another row in the middle of the table, you click the numeric row number to select the entire row. Then right-click and choose Add Row Above or Add Row Below from the secondary menu.

If you simply want to wrap text in a specific selected row's cell, you select the Text tab while the table is selected and enable Wrap text in a cell. That should be enabled by default. This will increase the row height of all cells on that specific row varied by the amount of text that you wrap, and does not affect other row heights.