I can't provide you with a complete solution (there are many) but notice one thing - if you add a link (a tag) the browser takes this thing and say something like "oh, I have to load this (different) page".
And that causes the browser to make a GET to the server for this new page.
So the first thing to do is - replace the link with a button (hyperlink button), which calls a method on the server.
Next is to change the content.
The easiest approach for me seems to build a usercontrol which renders the thing you want.
The masterpage concept is different to a "frame concept". While in a frame scenario you can tell every frame what to use (which content to show) a masterpage is something opposite.
You have a page "SecondPage.aspx" - which uses the same masterpage as "FirstPage.aspx".
With frames you would do something like ContentFrame.src="http://...." while the "framework around" (masterpage content) would stay the same.
With a masterpage you now ask the server to bring up SecondPage.aspx. The page loads (on server) and ASP.NET detects - ah this page uses a masterpage. So the server "surrounds" the content of SecondPage.aspx with the elements from the masterpage.
Then the whole thing gets sent to your browser.
Even if not "technical perfect" think of a masterpage as something like a backpack which every single page carries.
Or think of it as clothes - this makes every page you create "look the same" - BUT - like an actor in the theater - clothes are not the stage - they are the things actors wear. And calling a different actor to enter the stage means that the actor has to move there.
Or here - a page is loaded. No matter what the actor wears - he has to enter the stage -- the page gets loaded.
In other words - using masterpage makes all your pages look uniform - because the "wear" the same uniform - but a page is still a single unit retrieved by loading it.
A solution for this is to have the "actor" carrying a "picture" - in other words you stay on the same page, but change the content of some part of it.
This can be done by (for an example) using RadPageView / TabStrip or with a user control which renders it's content depending on parameters, or..... (there are a lot of ways).
What to use depends on the solution you try to build - for (lets say) a few simple pages the PageView like shown here: