Grenadiers were originally the soldiers in charge of throwing grenades (hence the name). They were chosen for being tall, muscular, and fearless - not only did they need the strength to throw grenades far enough to actually injure the enemy - they had to be brave (or stupid) enough to stay in the front lines of battle to do it.
Grenadiers were also called upon to be fast, effective assault troops - helping with things like storming breaches during sieges. (You know, all the standard easy jobs that are likely to get you killed.)
By the mid-18th century grenades had fallen out of favour, but grenadiers were still regarded as elite regiments, called on to guard the right flank of their battalions - and still to do all the other dangerous work they'd gotten a reputation for.
And interesting note on the hats - they were made high and narrow so that as not to get in the way while you were throwing grenades (which is a good feature to have). Eventually, the miter hat became part of the standard dress uniform in numerous armies.
It's also possible grenadiers just really dig the idea of looking like the Pope. History's mute on that point.