This phrase means, in essence, that one can more successfully win others over with a little kindness than a great deal of something less sweet: guilt, anger, or sarcasm, for example.  If you want someone to do you a favor, especially something that you'd have to convince them of, something they might not really want to do, it is much easier to get them to do it if you are kind and ask politely and maybe even offer to do a little something for them in return.  On the other hand, if you begin by explaining that they really owe you or should help you because it's only fair or that you'll be really angry with them if they don't, you stand a much smaller chance of them wanting to help you out.  It's a pretty simple idea that, in my experience, seems completely true: if you are nice, people will like you and want to help you; if you are "tart" or angry or selfish or bitter, then people won't like you and they won't want to help you.

What this saying, repeated by Benjamin Franklin, means is that it is easier to get people to do what you want by being nice than by being mean.

When Franklin uses the phrase "tart words" he means words that are angry.  He says that if you use angry words when talking to people they will not want to be your friend.

Instead, he says, you have to use honey.  When he says this, he mans that you have to use nice ("sweet") words and actions.  If you act with a little kindness, it will be more effective than doing or saying many angry things (the gallon of vinegar).

In this saying, Franklin is using sour things as a metaphor for anger and sweet things as a metaphor for kindness and nice actions.  He is saying that nice words and actions are more effective than angry ones.