As regards to your question " Why would someone want to convert columns into rows (ie pivoting)? Which problem does it solve?"
It's not a matter of "how data looks visually" but rather "how it is to be organized and handled". Namely:
In most of the databases, rows represent "Records - (entity, event, etc)" and columns "Fields (attributes of that entity)". For instance the below is a typical DB representation of data;
Person | Birth | Height |
-------------------------------
John | 1980 | 1.82 |
Smith | 1987 | 2.02 |
That means; each column represents a particular attribute of "persons" and when you select a particular column, you get that particular attribute of ALL the people. This way, a column is a "Dimension" and all values have the same unit (data type), all are dates, all are lengths, etc.
In financial modeling, however, it is just much convenient to represent data the other way around. For instance a typical in "Monthly Cash Flow" table looks like this.
Item | Jan | Feb |
-----------------------------
Sales | $100 | $150 |
Tax | -$50 | -$15 |
Note that this kind tabulation in a spreadsheet does NOT adhere database formats, column heading are time, but the values in the columns are monetary values, CONFLICT, we can't do calculations with this columns.
If we PIVOT this table, it becomes
Date | Sales | Tax |
------------------------------
Jan | $100 | -$50 |
Feb | $150 | -$15 |
Now, column dimensions (heading) and the data inside them are consistent.
Date column have all dates, and the others have all $ figures. We can take a column and do vector operations with it.
This is one problem that pivoting solves.