Actually they used to show the exact number; however this resulted in people comparing (and really, competing) their profiles solely based on this number, no matter what those connections were really like. Some sites, like TopLinked, even created public leaderboards to showcase profiles with the highest amount of contacts – and offered a $10/month service to help you boost that number...
Obviously, LinkedIn didn't like this. As Chris Nodder writes in his book, Evil by Design:
This again appealed to users' competitive nature but was completely counter
to LinkedIn's mission to create a "professional network of trusted contacts."
LinkedIn claims that "Connecting to someone on LinkedIn implies that you know
them well" and that makes business sense. LinkedIn makes its money from analyzing
the connected nature of individuals to determine their traits and target
advertising. If the connections are random, there is no money to be made. It also
makes money from upselling users on premium features such as wider searches
(beyond third-level connections) and the ability to send "InMails" to people outside their network. Again, if an individual has a massive network, this reduces the need for these services. Steven Burda, a highly linked individual, claims he can reach 97 percent of LinkedIn members via his third-level connections.
When LinkedIn changed users' profiles to show a generic label of "500+
Connections" rather than a true figure, TopLinked could no longer maintain
its leader board.
So there you have it – they do this to guard themselves and their users against services looking to exploit competitive nature and pride.