awk, like grep, in its default invocation, operates on
lines (records, separated by newlines), and fields (by
default separated by whitespace other than newlines).
What the awk snippet above does is to re-define records
(RS=record separator, ORS = output RS) and fields (FS=
field separator, OFS=output FS) to be the following
RS = "\n\n" = two subsequent new lines (in other words: an empty line)
FS = "\n"
So, an empty line defines a boundary between two records
(which is what you referred to as a paragraph).
Now, with the BEGIN statement awk has its behaviour
for all records and fields defined as you need them,
with NF >=4 it will print any record that has more than
4 fields (i.e., a paragraph with 4 or more lines).