They were like puzzle pieces that slipped from my hands the moment I picked them [up].
Is the up necessary? Why or why not?
Actually to pick something and to pick something up have two distinct meanings:
In your sentences you are specifically referring to 'pieces that slipped from my hands' so pick up is the correct verb to use.
'up' describes completeness. When you use 'up' it shows you are selecting something and lifting it up whereas when you use 'pick' alone it shows you are just selecting which seems sort of incomplete.
'to pick', and 'to pick up' are two different verbs. 'to pick up' is what's called a phrasal verb, formed by modifying the simple verb 'to pick'.
But they have different meanings! 'pick up' is to take something and raise it, or to collect someone in your car. 'I picked up the litter'. 'I picked her up from the station.'
'pick' is to choose. 'I picked my favourite'.
So without the 'up', your sentence means 'They slipped from my hands the moment I selected them'.
Presumably once upon a time, the meanings were much closer, and 'pick up' really was 'pick' with an extra adverb. But not in modern English.
I can't make this a "comment" due to lack of rep.
@John Lawrence Aspden
In a phrasal verb like "pick up", the particle "up" is not "an extra preposition". It happens that the word "up" can be used as a preposition, but it can also be an adverb:
"Look up! Look down! Look left! Look right!" The word "up" in this case is an adverb modifying the verb "look". (Note that this does not mean at all the same thing as the transitive phrasal verb "look up" as in "Look that up in your Funk & Wagnalls".)
However "Look up the street at the woman in the red dress." is using "up" as a preposition.
This happens to be a bit of a peeve of mine due to people using it as some sort of "proof" that the rule against ending a sentence with a preposition is wrong (citing Winston Churchill as some sort of authority). When you use "up" as a particle in a phrasal verb, it isn't being used as a preposition at all, so it does not in any way invalidate the idea that a preposition ought to be pre-positioned before its object.