Etymology/meaning of character 绍 and its relation to 介绍
Just my opinion after a quick look!

1 - Why does 绍 have a silk in it?

http://www.zdic.net/z/21/xs/7ECD.htm

绍: (1) (形声。从糸( mì),召声。本义:继承;紧密连续)

I would say, 糸( mì) is not really silk, just a figurative thread, band or bond.

Doesn't a thread, for example, in chinese.stackexchange.com, carry on? The idea is not new.

形声: 召声 is the sound of 绍.

2 - What's the meaning of 绍 in 介绍?

http://www.zdic.net/z/15/js/4ECB.htm

介：在两者中间: 介于两者之间。between both sides (friendship)
绍: 连续 go on without stopping

introduce: the etymology is not clear, but may be:
intro- means in (between)
ducere: lead (the one to the other perhaps)

Overall: 介绍： between (2 or more people) tie a bond which (hopefully) lasts.