There are a few ways that you can deal with this multiple values for a single record. You could have another table that holds these values and has a column to relate to the main table with this info. This is probably the best way to go so that you can keep all of the data intact and easily retrievable.
You can also modify your existing page to allow for more than one value to be inserted into the column of your db based on the multi select from the form. This way though will leave you with a column full of value for a particular record and may not be the easiest way to retrieve results.
If you would like to have it work the way it is now you would need to do two things:
First, you will need to find the select input on the page and add  to the end of it's name and id. This will force the php code to look at the data as an array of data, which it is.
If you just changed this you would seen an entry in your db that just said array. To break out the individual pieces you will need to add in an implode function around the value for this select list. When you add in the implode you will need to decide what character you would like to use as the separator.
In the code on the page this will look like this for your update:
implode(", ", $_POST["your select list"])
This will make the array of user selections into one string with the values separated by a comma and space (, ). This implode function would be added to the source code inside of your update or insert server behavior. If you post back with the page in question in a zip archive I can show you where these changes would need to be made.