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Note that "patterns" may be one or more, separated by newline, so grep $'a\n' infile is the equivalent of

grep 'a
' infile

or grep -e 'a' -e '' infile, so you are providing the pattern a (with no matches in this case) and then a null pattern '' which matches every line.  It's also mentioned in the GNU Grep Manual:

11. Why does the empty pattern match every input line?

    The ‘grep’ command searches for lines that contain strings that match a pattern.  Every line contains the empty string, so an empty pattern causes ‘grep’ to find a match on each line.

grep $'a\nb' is the equivalent of

grep 'a
b' infile

or grep -e 'a' -e 'b' infile.  Since there are no matches in your input, nothing is printed.

You ask

Since '' matches every line then why doesn't -o print anything?

The GNU man page says (emphasis mine):

-o, --only-matching

Print only the matched (non-empty) parts of a matching line…

I don't have access to OSX right now but I'm pretty sure their man page mentions this too…  It's also in the GNU Grep Manual.  Since the matched part of the matching lines is empty, nothing is printed.

grep accepts a list of "patterns" (i.e. one or more), separated by newline, so grep $'a\n' infile is the equivalent of

grep 'a
' infile

or grep -e 'a' -e '' infile, so you are providing the pattern a (with no matches in this case) and then a null pattern '' which matches every line.  It's also mentioned in the GNU Grep Manual:

11. Why does the empty pattern match every input line?

    The ‘grep’ command searches for lines that contain strings that match a pattern.  Every line contains the empty string, so an empty pattern causes ‘grep’ to find a match on each line.

grep $'a\nb' is the equivalent of

grep 'a
b' infile

or grep -e 'a' -e 'b' infile.  Since there are no matches in your input, nothing is printed.

You ask

Since '' matches every line then why doesn't -o print anything?

The GNU man page says (emphasis mine):

-o, --only-matching

Print only the matched (non-empty) parts of a matching line…

I don't have access to OSX right now but I'm pretty sure their man page mentions this too…  It's also in the GNU Grep Manual.  Since the matched part of the matching lines is empty, nothing is printed.

Note that "patterns" may be one or more, separated by newline so grep $'a\n' infile is the equivalent of

grep 'a
' infile

or grep -e 'a' -e '' infile so you are providing the pattern a (with no matches in this case) and then a null pattern '' which matches every line.  It's also mentioned in the GNU info documentation:

  11. Why does the empty pattern match every input line?

     The ‘grep’ command searches for lines that contain strings that
     match a pattern.  Every line contains the empty string, so an empty
     pattern causes ‘grep’ to find a match on each line.

grep $'a\nb' is the equivalent of

grep 'a
b' infile

or grep -e 'a' -e 'b' infile and since there are no matches in your input, nothing is printed.

Note that "patterns" may be one or more, separated by newline, so grep $'a\n' infile is the equivalent of

grep 'a
' infile

or grep -e 'a' -e '' infile, so you are providing the pattern a (with no matches in this case) and then a null pattern '' which matches every line.   It's also mentioned in the GNU Grep Manual:

11. Why does the empty pattern match every input line?

    The ‘grep’ command searches for lines that contain strings that match a pattern.  Every line contains the empty string, so an empty pattern causes ‘grep’ to find a match on each line.

grep $'a\nb' is the equivalent of

grep 'a
b' infile

or grep -e 'a' -e 'b' infile.  Since there are no matches in your input, nothing is printed.

You ask

Since '' matches every line then why doesn't -o print anything?

The GNU man page says (emphasis mine):

-o, --only-matching

Print only the matched (non-empty) parts of a matching line…

I don't have access to OSX right now but I'm pretty sure their man page mentions this too…  It's also in the GNU Grep Manual.  Since the matched part of the matching lines is empty, nothing is printed.

Note that "patterns" may be one or more, separated by newline so grep $'a\n' infile is the equivalent of

grep 'a
' infile

or grep -e 'a' -e '' infile so you are providing the pattern a (with no matches in this case) and then a null pattern '' which matches every line.

grep $'a\nb' is the equivalent of

grep 'a
b' infile

or grep -e 'a' -e 'b' infile and since there are no matches in your input, nothing is printed.

Note that "patterns" may be one or more, separated by newline so grep $'a\n' infile is the equivalent of

grep 'a
' infile

or grep -e 'a' -e '' infile so you are providing the pattern a (with no matches in this case) and then a null pattern '' which matches every line. It's also mentioned in the GNU info documentation:

  11. Why does the empty pattern match every input line?

     The ‘grep’ command searches for lines that contain strings that
     match a pattern.  Every line contains the empty string, so an empty
     pattern causes ‘grep’ to find a match on each line.

grep $'a\nb' is the equivalent of

grep 'a
b' infile

or grep -e 'a' -e 'b' infile and since there are no matches in your input, nothing is printed.