Ah, well, don't want to disappoint you here, Neil!!
As DomRUK says, they are a kinaesthetic connection between the sound and a physical movement.
As Neil says, each one has a different shape which help to express the function of the note within the scale.
If you use them regularly then the vocal pitch/interval becomes so connected with the physical hand movement that it becomes almost impossible to show one handsign and sing a different pitch.
They are also very helpful as a form of sight-singing - the hand shape recalls the sound, so the teacher can show a phrase or motif with handsigns and the students can sing it in solfa immediately - it's like magic!
The teacher can also show a phrase, motif or whole song with handsigns for the student to sing with inner hearing and then sing aloud, thus aiding memory as well as pitch. (If we had a webcam link here I could have HANDSIGNED Bobby Shafto for all to recognise with 'thinking voice'!!
Bagpuss noted the other day in my class that we sang something and the intonation wasn't wonderful - we repeated it with handsigns and it went bang in tune
A teacher can also use both hands - the class is divided into two and each group reads from one hand. This is fantastic intonation and interval practice.
Ah - I just thought - I did say earlier in my answer that it is very difficult to sing one pitch and handsign another - but there is a specific instance when we would definitely ask students to do this: sing a song (which works in canon) in solfa and HANDSIGN the canon - not purely a co-ordination exercise but develops the harmonic hearing too.
Also for similar reasons, in a two-part piece - sing one line while handsigning the other!
This is by no means an exhaustive list of the uses of handsigns but I hope it gives a little insight