Why does Bilbo sing songs to the spiders?
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At this point Bilbo has his ring on and is invisible. He sings to disorient them as well as frustrate them,
"Here I am, naughty little fly; / you are fat and lazy. / You cannot trap me, though you try, / in your cobwebs crazy."
So he can let them go away
As,the dwarvs were in trouble by the spiders Bilbo was helping the dwarves by distracting the spiders and scaring them.He was invisible and throwing stones at different directions.He was also singing by going at different places beacause his plan was that whenever spiders got distracted by Bilibo's singing and went to other places he would cut the spider webs by which the dwarves were being hanged and save the dwarves