What and where is the connection between 压岁钱 and the character 岁 that appears in it?
"压岁" and "压祟" have the same pronunciation (yā suì). The meaning of "压祟" is exactly what you have mentioned here: "suppress or put down the evil spirit". "压岁钱" is actually a homonym of "压祟钱".  It's a usage passed down from ancient time.