Why is there no 'the' before the numbers in expressions like 'finish in first place'? Even Grammarly underlines it. What is the grammatical explanation?
"In first place", is a fixed phrase, meaning before all other finishers in a contest. The word "first" acts as a determiner, uniquely identifying the particular "place" involved. It may be contrasted with "second place" and "third place", also usually without an additional determiner.
In some cases, when "first-place" is used as an adjective, it may be preceded by "the", for example, "the first-place finisher", but then the word "the" belongs to the noun "finisher".
"In the first place..." has a different meaning, and is defined in dictionaries, for example,
Merriam-Webster "in the first place"
1 —used at the end of a sentence to indicate what was true or what should have been done at the beginning of a situation
We should never have gone there in the first place.I didn't care much for the job in the first place.
2 —used when listing the most important parts of something or the most important reasons for something
I'm not going to tell you because, in the first place, it's none of your business, and in the second place, you would tell everyone else.
In that sense, "the first place" may be contrasted with "the second place" or "secondly".