What does PARTITION BY 1 mean?

It is an unusual use of PARTITION BY. What it does is put everything into the same partition so that if the query returns 123 rows altogether, then the value of rows_in_result on each row will be 123 (as its alias implies).

It is therefore equivalent to the more concise:

COUNT(*) OVER ()

Databases are quite free to add restrictions to the OVER() clause. Sometimes, either PARTITION BY [...] and/or ORDER BY [...] are mandatory clauses, depending on the aggregate function. PARTITION BY 1 may just be a dummy clause used for syntax integrity. The following two are usually equivalent:

[aggregate function] OVER ()[aggregate function] OVER (PARTITION BY 1)

Note, though, that Sybase SQL Anywhere and CUBRID interpret this 1 as being a column index reference, similar to what is possible in the ORDER BY [...] clause. This might appear to be a bit surprising as it imposes an evaluation order to the query's projection. In your case, this would then mean that the following are equivalent

COUNT(*) OVER (PARTITION BY 1)COUNT(*) OVER (PARTITION BY col_a)

This curious deviation from other databases' interpretation allows for referencing more complex grouping expressions.