Selecting rows conflicting values in WHERE clause
The answer: as ypercube said, the solution is:

adding parentheses around the OR conditions  

like so:  

SELECT name,url FROM additions WHERE enabled = 1 AND picture = 1 AND (name LIKE '%value%' OR tags LIKE '%value%' OR subtext LIKE '%value%');


With the parentheses, AND will take precedence and the entire section in the LIKE value comparison will have to be true together with the other evaluations, like enabled = 1 and picture = 1. Without the parentheses the OR statements are evaluated individually (and if one of them happens to be true, a row will be selected).