@Dwabii, How you inspire confidence in both the design and execution is a bit of salesmanship really. You just need to be convincing that you have the solutions to the design/tech questions they will ask ready to fire. The prototype should be in working order far enough that they can really see the “light at the end of the tunnel” - which really varies from game to game. Some game designs it is VERY clear from an early stage where the game will go. Others it isnt clear how a game plays till its launch day 
Once you have the stage at which you can pitch effectively in hand, you then need to have the answers ready for them and a good sales pitch as to what you want / need from them and why that this partnership is beneficial to both sides.
Again, really, the days of publishers is setting at the moment. There’s SOME things they can still provide, but ironically the two things people tend to want them for are money and PR work. The first they are typically unwilling to do much on for advances vs. royalty… maybe some if you really have a good pitch but I have found getting money from a publisher up front is often a disastrous process. PR work is important, but the reality is for 1-3k/month you can hire a professional game PR person.
Some of the things publishers CAN provide (in addition to the sometimes money and the sometimes PR help) is access to conferences you may not have the budget/time to reach (PAX, E3, GDC, Gamescom etc - get where they are going in writing and ensure you are included). Also bundle deals & sale events on Steam which you may have trouble negotiating as a single game developer. Also, sometimes, other services ranging from translation/localization, retail, and maybe most vital of all, console port opportunities.
Really though money and PR are the two less valuable items they offer. The first because it is hard to get and the second because the first can get it for you