You have to have a contract with Twitter. Which they did. Until Twitter decided not to continue to renew their contract with Google.
Now Twitter is complaining that Google isn't serving their results prominently. I honestly cannot begin to comprehend the level of cognitive dissonance needed to whine about this.
I doubt that Twitter just told Google "Sorry, we don't want to do business with you." Most likely either Google wanted a better deal and couldn't get one, or Twitter raised their rates. Which is kind of what the Twitter rep is saying when they state that they continue to work with lots of other companies of various sizes.
I'm not entirely sure that was the terms of the agreement. If I remember correctly, Google showed those tweets at the top of the SERP. How is much the #1 spot on all Google SERPs worth?
I have no idea what you have to do to get access to it, though.