Why was he doing this? He just wished that someone would tell him why he was doing this. Running after Sherlock. Running after Sherlock who was running after a taxi which may or may not be housing a murderer. This was insane!
Although, John had the feeling that Sherlock would run headfirst into traffic to stop this taxi if John wasn't there to stop him. It was still a matter of why John was apologizing to everyone that Sherlock was running down, or everyone that was almost running him down.
This was insane! He hadn't done anything this crazy in a long time, and that was saying something since he had invaded Afghanistan!
Oh, that was a far, high jump- Just do it, John! Keep up with Sherlock!
There was a sense of danger, definitely, but there was also a sense of... a thrill, of fun, of excitement. John missed this stuff. John really missed this stuff.
"Welcome to London."
John was chuckling, even though he couldn't breathe, because Sherlock was just plain mad and this was all just plain mad, but goodness, did he enjoy it.
"Come on, John!" Thus, the beginning of a inseparable friendship.