Why arrival process of packets at a switch is not a Poisson Process?
"new flow arrivals" means "arrivals of new flows".  A flow is a TCP connection (roughly); each individual TCP connection is a separate "flow".  So, this is talking about new TCP connections, and the rate/time at which the server receives new TCP connections.

Contrarily to the statement you quoted, web requests won't necessarily be Poisson.  There are many factors that can cause web requests to be other-than-Poisson-distributed.  For instance, the Olympics web site receives a lot more traffic when the Olympics events are happening than when they aren't.  Many web sites receive more traffic during the time of day when their primary user population is awake than when they're not.  In addition, visiting one page on a site typically triggers many requests for other related resources (e.g., images that are loaded on that page).

However, during any sufficiently short time period, if their user base is sufficiently large, the arrival process for a particular type of web request might well be approximately Poisson.

See, e.g., https://en.wikipedia.org/wiki/Poisson_distribution#Occurrence.