It is just a way of expressing a great deal of happiness. Larks soar in the sky, and have a beautiful song. It appears sometimes that they are moving and singing for nothing but the joy of it.
I have two theories:
A mudlark is a bird that could be found foraging on the banks of the River Thames in Victorian London, when the river was heavily polluted. Presumably they could have looked...
The use of "as" in the phrase "happy as a lark" incorporates the use of a simile. In addition, lark is typically used as a symbol for hope, happiness and good fortune.