I'm not entirely sure I understand how this menu works.

How does one go about changing the options in the menu? I know that up/down changes the selection, but how would I say change my tire types in the menu, whilst in a race?

Using left/right arrow keys

Quote from r4ptor :

Using left/right arrow keys

Not working for me

You are in singleplayer mode and not using latest patch then, eh? With older patch it only works online, but with latest it also works offline

check if cursor keys are assigned for any other action in lfs (steering?) - remove them there.

Quote from -Willi- :

check if cursor keys are assigned for any other action in lfs (steering?) - remove them there.

Nope :\

And i'm running W.

If arrows not working, try Shift + arrows.

Quote from sinkoman :

Nope :\

And i'm running W.

If you get the latest Test Patch here :

http://www.lfsforum.net/showthread.php?t=24646

Those buttons now work in single player mode.

One of my team mates also has to use shift+arrows, what can he do to use arrows?

Quote from Nobo :

One of my team mates also has to use shift+arrows, what can he do to use arrows?

Unassign the arrow keys from whatever they are assigned to in Options... Controls.