Because

I'm a nice person - why can't I sit in here at one of the tables way over there and use my computer, have a conversation, sit and read?

I'm sure you are a nice person and not a pedo or anything, but this is the Children's Room and it is for children and their caregivers. 

But there's no one here right now -

This is one small part of this library.  It takes up maybe one sixth of the public space.  The Teen Center is even smaller.  And all the rest of that belongs to adults.  Why don't you explore?

I've already walked all the way down here, you know.

And we are so proud of you.  Now show us how you can walk all the way back to the front (which you'll have to do to leave anyway), hop on the elevator, and walk down to the rotunda where there are tables and chairs and you can spread out.  Also, you won't be intimidating the children.  It is unwitting, I know, but just the presence of "strange" adults will frighten some children away from a certain area - such as those nice tables over by the windows where the manipulatives and crayons are.  And those couches by the puppet theatre?  Those are for parents to watch their children put on a puppet play or play at the duplo table. 

Most children are trained to be quiet around adults when they are doing adult things (not all, so I've noticed, but you get the idea).  If you are talking to a friend and aren't obviously a parent or grandparent with a child also playing, you are an impediment to play and, therefore, learning.  And even if they don't mind, won't their boisterous play interfere with your work or conversation?  If that is the case, I will ask you to leave rather than ask them to be quiet for your sake because this is their area.

By the way, the library is not a place to come and have a conversation.  If your main intent is to talk, either use one of the small conference rooms that are available on a first-come-first-serve basis ... or go somewhere else.  Yes, there are some ladies at the couches chatting away, but you know what?  They have children playing right there.  If they are letting the children get away with screaming bloody murder, I will have a separate chat with them.  The point is that they remembered to bring their children with them.  Next time why don't you?

[Now, the hard part is to put this in the tone of voice but reduce the verbiage to, "This is the Children's Room."  Practices Great Big Smile.]