A lookup table with a single value (StateAbbr) should contain two fields: StateId (Primary Key) as an autonumber type, and StateAbbr as a text type. Your Persons table should contain a number type field that will store the value held by the StateId field in the StateAbbr table. So,
The query used would be something like:
SELECT Persons.StateAbbr, States.StateAbbr FROM Persons, States WHERE Persons.StateAbbr = StatesAbbr.StateId;
Paste this statement into the SQL view of your query builder, then switch back to the design view to see the graphical representation.
This will return the actual state abbreviation code that you have assigned in the StateAbbr table.
Edit: I have included a screenshot of a quick sample I threw together based on what you asked, showing the two tables (design view), the relationship between them, and the SQL query that delivers the results I believe you are seeking. Modify these as you need to in order to make it work for you. This screenshot changes the query I provided above to better meet naming conventions for tables and queries.
NOTE: In the Persons table, you will create the State field using the Lookup wizard. Point it to the States table as its source, and only include the StateAbbr field. The PK in the States table will not be shown at all. When filling the data into the Persons table, you will notice that it is a drop-down box where you simply select the state you want.
Please let me know if you need me to clarify anything else.