Picture a net stretched under a whole country. Most of the time nobody thinks about it. Then someone loses a job, or falls ill, or gets too old to work, and the net is what stops the fall — money while they look for work, a doctor they can afford, a pension. The net is paid for by everyone's tax, so it belongs to everyone and everyone argues about it. Once you can see the net, the second use follows on its own: a country can be named after the net it has.
From the word’s history
The source records the phrase as a compound of welfare + state, which lines up with the picture: a state whose job includes keeping people well.
資料來源把它記成 welfare 加 state 的合成詞。字面就是「照顧人民過得好的國家」,正好對上那張網的畫面。
Wiktionary entry supplied in this packet (etymology_text: 'Compound of welfare + state.')