Richardson excels at scoring off-ball
Some players are talented offensive threats but need the ball in their hands to be effective. A guy like Josh Richardson knows where to be in offensive sets and does a good deal of moving without the ball to put himself in a position to be effective without dribbling around for 10 seconds.
A fair amount of Josh's scoring punch this season has come from catch-and-shoot threes. When his teammates aren't finding him wide open, he'll typically make a quick move to the hoop before throwing up a high-percentage shot in the paint or dishing the ball elsewhere. Richardson has also developed some considerable range on his jumper. When he's catching it from 30 feet out or more, he's more than comfortable letting it fly.
Richardson's ability to be an off-ball scorer is a good complement to his teammates' abilities. He's allowed Dejounte Murray to continue to be the focal point of the Spurs' attack and helped guys like Keldon Johnson and Devin Vassell progress further without needing to have the offense run through him. Since arriving in San Antonio, he's had the highest true-shooting percentage of his career and ranks in the 77th percentile of points per shot attempt, per Cleaning The Glass.
Locker Room Presence
I think the best way to sum up Josh Richardson's impact on the team as a whole is that he's just a great guy to have around your team. He's a seasoned vet and uplifting locker room presence, but he's also just a solid gold individual.
When the Spurs played the Thunder on March 16th, San Antonio was down one with 4.8 seconds left, when Lonnie Walker IV hit a clutch three to give them the last-second win. Lonnie later revealed that, in the timeout leading up to his shot, Josh Richardson told him he was going to hit the game-winner. "That might be my post-career plan, telling fortunes," said Richardson.
The fans of San Antonio already have a strong connection with J-Rich, and that's partly due to his long-time appreciation for the team's winning culture. When he first joined the team in February, he talked about how he had admired Gregg Popovich and the Spurs' brand of basketball since his college days at Tennessee.
Overall, Josh Richardson just seems to really enjoy being in San Antonio and it's been a great fit for both parties. When he's not on the court for the Spurs, he's attending a San Antonio FC match or staying active on Twitter, hyping up his teammates and interacting with fans (and oftentimes beating them at FIFA). He's said he enjoys the vibe of the city.
When you have a player that makes sense for your team basketball-wise and meshes with so many through his personality, it's a no-brainer to try to keep him around for as long as possible.