What does [FROM x, y] mean in Postgres?
The manual has detailed explanation for the comma in the FROM list in the chapter Table Expressions:

The FROM Clause derives a table from one or more other tables given in
a comma-separated table reference list.
FROM table_reference [, table_reference [, ...]]
A table reference can be a table name (possibly schema-qualified), or
a derived table such as a subquery, a JOIN construct, or complex
combinations of these. If more than one table reference is listed in
the FROM clause, the tables are cross-joined (that is, the Cartesian
product of their rows is formed; see below).

The fact that comma-separated table references have been defined in an earlier version of the SQL standard than explicit JOIN syntax does not make the comma wrong or outdated. Use explicit join syntax, where it's technically necessary (see below) or where it makes the query text clearer.
The manual again:

FROM T1 CROSS JOIN T2 is equivalent to FROM T1 INNER JOIN T2 ON TRUE
(see below). It is also equivalent to FROM T1, T2.

There are corner cases where both are not completely equivalent. The manual:

Note
This latter equivalence does not hold exactly when more than two
tables appear, because JOIN binds more tightly than comma. For example
FROM T1 CROSS JOIN T2 INNER JOIN T3 ON condition is not the same as
FROM T1, T2 INNER JOIN T3 ON condition because the condition can
reference T1 in the first case but not the second.

This related question demonstrates the relevance of the difference:

“invalid reference to FROM-clause entry for table” in Postgres query

Basically, your observation is exactly right:

it seems to me that it's declaring a variable called query so it can use it multiple times.

Any function can be used as &quot;table function&quot; in the FROM list.
And function parameters can reference columns from all tables to the left of the function, because the notation:
FROM apod, to_tsquery('neutrino|(dark &amp; matter)') query

... is really equivalent to:
FROM apod CROSS JOIN LATERAL to_tsquery('neutrino|(dark &amp; matter)') AS query

The manual on LATERAL queries:

Table functions appearing in FROM can also be preceded by the key word
LATERAL, but for functions the key word is optional; the function's
arguments can contain references to columns provided by preceding FROM
items in any case.

Bold emphasis mine.
The keyword AS is completely optional noise before table aliases (as opposed to column aliases, where it's recommended not to omit AS to avoid possible ambiguities). See:

Date column arithmetic in PostgreSQL query