When I use mysqli->real_escape_string and add a string value to the db such as "who's", it adds the string without modifying the quote in any way. When I check the db, the value within it is "who's".
When I do not use mysqli->real_escape_string and add a string with a single quote like "who's", it just doesn't get added to the database. It is nowhere to be found.
This isn't normal, is it?
Yes, it is. It's a little difficult to answer this question in any more detailed a way than that.
real_escape_string() only escapes the data so it can be safely used in a query, it doesn't modify it in any way. When you don't escape it, the query has a syntax error in it so it fails - and the data is not inserted.
However the safest way to escape your data is to use prepared statements.