The MySQL documentation says the following:

If you want to change the table default character set and all character columns (CHAR, VARCHAR, TEXT) to a new character set, use a statement like this:

ALTER TABLE tbl_name CONVERT TO CHARACTER SET charset_name;

Warning: The preceding operation converts column values between the character sets. This is not what you want if you have a column in one character set (like latin1) but the stored values actually use some other, incompatible character set (like utf8). In this case, you have to do the following for each such column:

ALTER TABLE t1 CHANGE c1 c1 BLOB;
ALTER TABLE t1 CHANGE c1 c1 TEXT CHARACTER SET utf8;

The reason this works is that there is no conversion when you convert to or from BLOB columns.

If you specify CONVERT TO CHARACTER SET binary, the CHAR, VARCHAR, and TEXT columns are converted to their corresponding binary string types (BINARY, VARBINARY, BLOB). This means that the columns no longer will have a character set and a subsequent CONVERT TO operation will not apply to them.
I don't understand the part in bold. How can a column have one character set but the stored values use another character set? I'm new to this, so based on the little I know it really confuses me.

Perhaps upon knowing that I'll better understand their proposed solution.

Thanks.