How can I tell my advisor that I'm not going to work for free?

I would suggest that you repeat that you did not win the fellowship and ask if he has other funding available, "because I am still very interested in the work, but will need to find paid work instead if I can't get a stipend for this collaboration."

I think you should act as soon as possible. There are other sources of funding, and there may still be time to get hold of one.

So, in that light, just send an email saying that you can't afford to work for free. Add that you thought it was clear from your previous discussions, so offer your apologies. And, finally, say that you are willing to try to apply for other grants.

I think the famous 'Let the truth set you free' applies here. If I were you I would handle the whole situation as following:

Meetings: I would have regular meeting with your supervisor to keep the conversation going and get the feedback from your supervisor.

Mentioning The Collaboration: Then, in one meeting I would say to him/her that I can not do the collaboration as I need to find a paid job. I would say this face to face, and not through email; so he/she would get the whole thing without any false judgement.