With this. what does "estimate of root" mean? actually, what does everything on that page mean?

Let's look at the first one.  We're interested in approximating , the square root of some natural number.  Here we have n=5, so we want to approximate

A good way to get at least the integer part of the approximation is to sandwich n between two perfect squares, i.e. to find a natural number x such that

Once we do that we know

The squares go

We see n=5 is between 4 and 9, i.e. between and so we conclude

We might guess and then adjust our guess downward because 5 is a lot closer to 4 than it is to 9.  They went with 2.2, good enough I suppose.

I'll do a couple of more.   We see from our  list

The 75 is closer to 81 than 64, so let's bump up 8.5 to say

OK, one more, n=17

17 is real close to 16 so instead of 4.5 let's go with