Transition expenses would be many things, basically the operating expenses of him and his staff from now to Jan 1. For example, all of the people he is bringing on like Lina Khan, need salaries, office space, transportation costs, and the other running expenses that are needed before he has the key to city hall.

I'm sure that there will also be some of the money for parties and celebrations, for example I just saw a post of his volunteer appreciation party.

This campaign is to avoid such large bribes that you bring up. He wants to fund the transition with individual donations rather than have Doordash and AIPAC foot the bill as most do.