Why do Pros skill an ability so late?

Solution 1:

The main point is flexibility.

but sometimes it's just a habit...

In case you are invading or you get invaded a skill might be better than another.

Example:

You normally skill Amumus E because of the faster jungle clear and sustain.

But when you're getting invaded Q might secure a kill because of the stun and higher damage.

but sometimes it's just a habit...

Solution 2:

DropDeadSander covered the general case well, there are also specific champions where you might leave picking a skill even later though.

For example with Tristana I usually skill Explosive Shot first but I don't even always skill it immediately I reach lane unless I see a chance to harass. This is both because as soon as I have it the passive starts pushing and it means I can pick Rocket Jump in an emergency.

In lane-swap situations pros have been known to reach level 3 or 4 with Tristana while only having one point in rocket jump and nothing else because they want to freeze the lane and as soon as they pick up Explosive Shot the passive starts pushing the lane out rapidly. Those sort of situations are very rare in normal play though so I wouldn't expect to do this yourself, generally as soon as you actually start farming in lane you want to select an ability.