You could either use a column link to select the record and navigate to another page, or a radio button and a page button/link to do it. I'll demonstrate both using a simple report on the DEPT table.
Method 1: radio button
For the radio button we can add an additional column to the report using the apex_item.radiogroup
function to create a radio button whose value is the DEPTNO:

By default, the HTML of the radiogroup will be escaped for security reasons, which is not what you want but illustrates what it is doing quite nicely:

We can fix that by changing the column property to "Standard Report Column":

Now we see:

Clicking on the radio button on any row selects it and deselects the buttons on other rows.
To navigate to another page with the selected row we need a button to submit the page with a special request:


When clicked, that button will submit the page with a Request value of "SELECT" (the button name I chose). So we can write an on-submit page process to fire when the request is "SELECT", find out which radio button has been selected (if any) and save the selected DEPTNO to a hidden item called say P34_DEPTNO. We find out which button by looking at the APEX array apex_application.g_f01
which we chose by passing 1
as the first parameter to apex_item.radiogroup
:
if apex_application.g_f01.count > 0 then
:p34_deptno := apex_application.g_f01(1);
else
:p34_deptno := null;
end if;
Then we can define a branch that navigates to the new page if (a) request = 'SELECT' and (b) P34_DEPTNO is not null.

And that's it. Quite a lot of work, but if that's the requirement that will do it.
Method 2: column link
The simpler way is to dispense with the radio buttons and just make one of the report columns into a link:

This turns the column (I chose DNAME) into a link that navigates to the new page taking the selected DEPTNO value with it:

That's it! No hidden item, no button, no page process, no branch...