The top 大 (a person with arms outstretched) is the indexing Kangxi radical. In 奢 it sits above 者, hinting at someone stretching beyond proper bounds - the imagery behind the 'extravagant, excessive, lavish' meaning seen in 奢侈.
The bottom 者 supplies the sound, giving shē with a tone-and-initial drift from zhě. 者 is the phonetic in a family of related characters - 诸, 都, 暑, 著 - and here it pairs with 大 to picture a person who indulges to excess.
