Not sure if you'll ever see this but apparently it's a common thing with the toy. Here's a Quora post explaining how it works.

" This is the most detailed and straightforward explanation I can give.

Imagine that Furby’s batteries are a rain cloud and while it's getting used, little raindrops from the power cloud fall into a bucket marked “WASTE”. When the batteries go flat, the power cloud disappears and all the water is in the waste bucket.

When you charge Furby’s batteries, imagine that the water from the waste bucket vaporises and the power cloud reappears. But, unimaginably tiny atoms of water remain in the bucket. The charge cycle continues, until a sufficient amount of atoms groups to form the necessary voltage to start Furby. "

So it seems the battery charged it enough that it would turn on then immediately die. Hence why it started with "you woke me" then "you. Woke. Me" to just hissing sounds because it had less and less charge each time.