The internal editor Kile uses is Kate. Kate is a text editor, created for programmers, which incorporates the ability to read and highlight many different types of text files, among which: LaTeX and BibTeX.
To learn more about Kate and it's capabilities you can reference the Kate Handbook here. Kile users can start reading from the “Working with the Kate editor” chapter.
If the user would like to select text they have the following options:
Hold left mouse button and drag mouse to highlight text
Click once on a word move the cursor to a new area
Click twice on a word selects the whole word
Click three time selects the whole sentence
Holding the left mouse button and dragging the text you want to select automatically copies the selected text to the clipboard.
Holding shift and using the arrow keys allows a user to select portions of the source code in the editor window.